Vectors, matrices, eigenvalues and eigenvectors

Size: px
Start display at page:

Download "Vectors, matrices, eigenvalues and eigenvectors"

Transcription

1 Vectors, matrices, eigenvalues and eigenvectors 1

2 ( ) ( ) ( ) Scaling a vector: 0.5V = 0.5 = = ( ) ( ) ( ) ( ) Adding two vectors: V + W = + = = ( ) ( ) a b λa λb A scalar times a matrix: λ = c d λc λd ( ) ( ) ( ) a b x y a + x b + y A matrix plus a matrix: + = c d z w c + z d + w ( ) ( ) ( ) a b x y ax + bz ay + bw A matrix times a matrix: = c d z w cx + dz cy + dw 2

3 Hence the product of a matrix times a vector: ( ) ( ) a b x c d y = ( ) ax + by cx + dy This matrix transforms the vector into another vector: Complex scaling of vector b 0.25b y v= a b ( ) w=tv= ( ) v= 0.5a ( 0.25b ) Shearing of vector parallel to x-axis y b v= ( a b ) w=tv= ( ) v= ( a+0.2b b ) 0.5a a x a a+0.2b x 3

4 4

5 A system of linear equations: Can be written as: ( ) ( x y { x 2y = 5 2x + y = 10 ) ( ) 5 = 10 From the first we obtain x = 2y 5, which gives in the second: 2(2y 5)+y = 10 or 4y 10+y = 10 or 5y = 20 i.e. y = 4, and hence x = 3. Finally, for the matrix A = ( ) a b c d define det[a] = ad bc and tr[a] = a + d for the determinant and the trace. 5

6 Forest succession: Gray Birch Blackgum Red Maple Beech Gray Birch Blackgum Red Maple Beech For example, the fraction of Red Maple trees after 50 years would be 0.5 the fraction of Gray Birch trees, plus 0.25 the fraction of Blackgum trees, plus 0.55 the fraction of Red Maples, plus 0.03 the fraction of Beech trees. 6

7 Write table as a matrix: A = and define the current state of the forest as a vector, e.g., V 0 = ( ) After 50 years the next state of the forest is defined by: V 50 = AV 0 = ( ) which is a forest with 5% Gray Birch, 36% Blackgum, 50% Red Maple, and 9% Beech trees. 7

8 The next state of the forest is V 100 = AV 50 = ( ) After 100 intervals of 50 years, the state is V 5000 = A 100 V0, where A 100 = which is a matrix with identical columns. 8

9 Now consider an arbitrary vector V = (x y z w), where w = 1 x y z, and notice that A 100 V = x y z w = 0.005(x + y + z + w) 0.048(x + y + z + w) 0.085(x + y + z + w) = 0.866(x + y + z + w) ( ), the succession converges into climax state. This climax vector is an eigenvector of the matrix A! 9

10 Eigenvalue problem: Av = { ax + by = λx cx + dy = λy ( a b c d ) ( ) x y = λ ( ) x y or { (a λ)x + by = 0 cx + (d λ)y = 0 Multiply first with (d λ), and second with b: { (d λ)[(a λ)x + by] = 0 b[cx + (d λ)y] = 0 Subtract second from first: [(d λ)(a λ) bc]x = 0 Because x 0: (d λ)(a λ) bc = 0 10

11 Characteristic equation: Since A = λ 2 (a + d)λ + (ad bc) = 0 ( ) a b, this can be written as: c d λ 2 trλ + det = 0 Hence: λ 1,2 = tr ± tr 2 4 det 2 11

12 Numerical example: Av = ( ) ( ) 1 2 x 2 1 y = λ ( ) x y Characteristic equation: tr[a] = 2 and det[a] = 1 4 = 3 Hence: λ 1,2 = tr ± tr 2 4 det 2 = 2 ± λ 1 = 3 and λ 2 = 1 = 2 ±

13 Corresponding eigenvectors: { (a λ)x + by = 0 cx + (d λ)y = 0 or { y = λ a b x = λ d c x y First eigenvector λ 1 = 3, a = 1, b = 2, c = 2, d = 1: y = x = x and x = y = y hence v 1 = ( ) 1 1 Second eigenvector λ 2 = 1, a = 1, b = 2, c = 2, d = 1: y = x = x and x = 1 1 y = y hence v 2 2 = ( ) 1 1 We only need one of the two equations! 13

14 Indeed, general case for eigenvectors: { (a λ)x + by = 0 cx + (d λ)y = 0 First equation delivers: ( ) x y = ( ) b a λ Indeed, second equals zero (delivers characteristic equation): bc + (d λ)(a λ) = 0 ( ) ( ) 2 1 Thus, λ 1 = 3, a = 1, b = 2: v 1 = or v 2 1 = 1 ( ) ( ) 2 1 for λ 2 = 1, a = 1, b = 2: v 2 = or v = 1 14

15 Special case, diagonal matrix: A = ( ) a 0 : 0 d λ 1,2 = tr ± tr 2 4 det 2 = (a + d) ± (a + d) 2 4ad 2 λ 1,2 = (a + d) ± a 2 + 2ad + d 2 4ad 2 = (a + d) ± (a d) 2 2 λ 1 = a and λ 2 = d 15

16 Special case, diagonal matrix: A = ( ) a 0 : 0 d Eigenvectors: { (a λ)x + 0y = 0 0x + (d λ)y = 0 λ 1 = a gives (d a)y = 0 or y = 0, i.e., v 1 = λ 2 = d gives (a d)x = 0 or x = 0, i.e., v 2 = ( ) 1 0 ( )

17 Linear differential equations The solution of dx(t)/dt = ax(t) is x(t) = Ce at, where C = x(0). Check this: t Ce at = ace at = ax(t) Now two-dimensional systems: { dx/dt = f(x, y) dy/dt = g(x, y) where x(t) and y(t) are unknown functions of time t, and f and g are functions of x and y. 17

18 An example: { dx/dt = ax + by dy/dt = cx + dy and { dx/dt = 2x + y dy/dt = x 2y where x and y decay at a rate 2, and are converted into one another at a rate 1. In matrix notation: ( ) dx/dt dy/dt = ( ) ( ) a b x c d y 18

19 We claim that ( ) dx/dt dy/dt = ( ) ( ) a b x c d y has as a general solution: x(t) = C 1 x 1 e λ 1t + C 2 x 2 e λ 2t y(t) = C 1 y 1 e λ 1t + C 2 y 2 e λ 2t or ( ) x(t) y(t) ( ) ( ) x1 = C 1 e λ1t x2 + C y 2 e λ 2t 1 y 2 where λ 1,2 are eigenvalues and (x i y i ) are the corresponding eigenvectors of the matrix given above. Like x(t) = Ce at, this has only one steady state: (x, y) = (0, 0). 19

20 ( ) ( ) ( ) x(t) x1 Notice that the solutions = C y(t) 1 e y λ1t x2 + C 2 1 y 2 are a linear combination of the growth along the eigenvectors. e λ 2t Since x(t) and y(t) grow when λ 1,2 > 0 we obtain: a stable node when both λ 1,2 < 0 an unstable node when both λ 1,2 > 0 an (unstable) saddle point when λ 1 > 0 and λ 2 < 0 (or vice versa) When λ 1,2 are complex, i.e., λ 1,2 = α ± iβ, we obtain a stable spiral when the real part α < 0 an unstable spiral when the real part α > 0 a neutrally stable center point when the real part α = 0 20

21 ( ) dx/dt Example: dy/dt = ( ) ( ) a b x c d y = ( ) ( ) 2 1 x 1 2 y Since tr = 4 and det = 4 1 = 3 we obtain: λ 1,2 = 4 ± 16 12) 2 so λ 1 = 1 and λ 2 = 3. = 2 ± 1 Hence solutions tend to zero and (x, y) = (0, 0) is a stable node. To find the eigenvector v 1 we write: ( ) ( ) b 1 v 1 = = a λ 1 1 or v 1 ( 1 1 ) 21

22 For v 2 we write v 2 = ( b a λ 2 ) = ( ) 1 1 In combination this gives ( ) ( ) x(t) 1 = C y(t) 1 1 ( ) e t 1 + C 2 e 3t 1 or x(t) = C 1 e t C 2 e 3t y(t) = C 1 e t + C 2 e 3t The integration constants C 1 and C 2 can be solved from the initial condition: i.e., x(0) = C 1 C 2 and y(0) = C 1 + C 2. 22

23 Let s check this solution: x(t) = C 1 e t C 2 e 3t y(t) = C 1 e t + C 2 e 3t or dx dt = C 1e t + 3C 2 e 3t dy dt = C 1e t 3C 2 e 3t which should be equal to dx dt = 2x+y = 2(C 1e t C 2 e 3t )+C 1 e t +C 2 e 3t = C 1 e t +3C 2 e 3t dy dt = x 2y = C 1e t C 2 e 3t 2(C 1 e t +C 2 e 3t ) = C 1 e t 3C 2 e 3t 23

24 Remember that we wrote: f(x) f( x) + x f( x) (x x): f(x) x f( x) f( x) x x

25 The function f(x, y) = 3x x 2 2xy: x f(x, y) = 3 2x 2y and y f(x, y) = 2x and in the point f(1, 1) = 0: x f(x, y) = 1 and y f(x, y) = 2

26 Generally f(x, y) f( x, ȳ) + x f (x x) + y f (y ȳ) Or, after defining h x = x x and h y = y ȳ: f(x, y) = f( x + h x, ȳ + h y ) f( x, ȳ) + x f h x + y f h y Example: f(x, y) = 3x x 2 2xy, f(1, 1) = 0, x = 1, y = 2 f(1.25, 1.25) = = f(1.25, 1.25) =

27 Consider { dx/dt = f(x, y) dy/dt = g(x, y) close an equilibrium point at ( x, ȳ), i.e., f( x, ȳ) = g( x, ȳ) = 0 Linear approximation of f(x, y) close to the equilibrium: f(x, y) f( x, ȳ) + x f (x x) + y f (y ȳ) As f( x, ȳ) = 0 we obtain For g(x, y) this yields: f(x, y) x f (x x) + y f (y ȳ) g(x, y) x g (x x) + y g (y ȳ) 27

28 { dx/dt = f(x, y) dy/dt = g(x, y) became { dx/dt x f (x x) + y f (y ȳ) dy/dt x g (x x) + y g (y ȳ) Since the partial derivatives are merely the slopes of f(x, y) and g(x, y) at the point ( x, ȳ), they are constants that we can write as a = x f, b = y f, c = x g, d = y g Steady states x and ȳ are also constants, with derivatives zero: dx dt = dx dt d x d(x x) = and dy dt dt dt = dy dt dȳ d(y ȳ) = dt dt Hence { d(x x)/dt = a(x x) + b(y ȳ) d(y ȳ)/dt = c(x x) + d(y ȳ) 28

29 Changing variables to the distances h x = x x and h y = y ȳ: { dhx /dt = ah x + bh y dh y /dt = ch x + dh y having the solution ( ) hx (t) h y (t) ( ) ( ) x1 = C 1 e λ1t x2 + C y 2 e λ 2t 1 y 2 where λ 1,2 and (x i y i ) are the eigenvalues and corresponding eigenvectors of the Jacobi matrix ( ) ( ) x f J = y f a b = x g y g c d 29

30 Having the solution ( ) hx (t) h y (t) ( ) ( ) x1 = C 1 e λ1t x2 + C y 2 e λ 2t 1 y 2 The Return time is defined by the dominant eigenvalue: 1 T R = max(λ 1, λ 2 ) when λ 1,2 < 0. 30

31 Having we know that J = λ 1,2 = tr ± D 2 ( ) x f y f x g y g where = ( ) a b c d D = tr 2 4 det Observing that λ 1 + λ 2 = tr[j] and λ 1 λ 2 = det[j], the latter because 1 4 (tr + D)(tr D) = 1 4 (tr2 D) = 1 4 (tr2 tr det) = det we can classify steady states by just the trace and determinant of their Jacobi matrix. 31

32 stable node stable spiral det center saddle non stable spiral 1 D=0 2 non stable node tr λ 1,2 = tr ± D 2 D = tr 2 4 det λ 1 + λ 2 = tr λ 1 λ 2 = det 1. if det < 0 then D > 0: λ 1,2 are real with unequal sign: saddle 2. if det > 0, tr > 0 and D > 0 then λ 1,2 > 0: unstable node. 3. if det > 0, tr < 0 and D > 0 then λ 1,2 < 0: stable node. 4. if det > 0, tr > 0 and D < 0 then λ 1,2 > 0: unstable spiral. 5. if det > 0, tr < 0 and D < 0 then λ 1,2 > 0: stable spiral.

33 Graphical Jacobian: use the signs only J = x f x g f( x + h, ȳ) h g( x + h, ȳ) h y f y g f( x, ȳ + h) h g( x, ȳ + h) = h α β γ δ with tr[j] = α + δ and det[j] = αδ βγ. If tr < 0 and det > 0 the state will be stable. 33

34 y y y (x,y+h) (x,y) x (x,y) (x+h,y) x (x,y) x a b c J = x f x g f( x + h, ȳ) h g( x + h, ȳ) h y f y g f( x, ȳ + h) h g( x, ȳ + h) = h + 34

35 The Graphical Jacobian of the Lotka Volterra model: N a c ( ) α β γ 0 e d R a b tr(j) = α < 0 and det(j) = βγ > 0

36 Finally: The full Jacobian of the Lotka-Volterra model: dr dt = ar br2 crn dn dt = drn en Nullclines: R = 0, N = a br c and N = 0, R = e d Steady states: (R, N) = (0, 0), (R, N) = (a/b, 0) and (R, N) = ( ) e da eb, d dc 36

37 Steady states of the Lotka-Volterra model N a c ( ) e d, da eb dc N = a br c e d R Note that da > eb a b 37

38 Lotka Volterra model: dr dt = ar br2 crn = f(r, N) and dn dt = drn en = g(r, N) New variables h R and h N define the distance to the steady state: dr dt = d( R + h R ) dt = dh R dt and dn dt = d( N + h N ) dt = dh N dt dh R dt dh N dt = f( R+h R, N+h N ) f( R, N)+ R f h R + N f h N ( R, N) ( R, N) = g( R+h R, N+h N ) g( R, N)+ R g h R + N g h N ( R, N) ( R, N) 38

39 Because f( R, N) = 0 and g( R, N) = 0: dh R = dt R f h R + N f h N ( R, N) ( R, N) dh N = dt R g h R + N g h N ( R, N) ( R, N) ( ) ( ) a 2b R c N c R? β J = d N d R = J = e +γ? The solution of the linear system has the form ( ) ( ) ( ) hr (t) R1 = C h N (t) 1 e λ1t R2 + C N 2 e λ 2t 1 N 2 where λ 1,2 are the eigenvalues of J and (R 1 N 1 ) and (R 2 N 2 ) the corresponding eigenvectors.

40 For ( R, N) = (0, 0) one finds: J = ( ) a 0 0 e with λ 1 = a > 0 and λ 2 = e < 0, i.e., a saddle point. For ( R, N) = (a/b, 0) one finds: ( a ac ) J = b 0 da eb b The eigenvalues are λ 1 = a < 0 and λ 2 = (da eb)/b. Since da > eb this is also a saddle point. 40

41 For ( R, N) ( ) = ed, da eb dc J = ( be d ce d da eb c 0 one obtains: ) = ( ) b R c R d N 0 = ( ) α β γ 0 Hence trj = b R < 0 and det J = cd R N > 0 which tells us that the non-trivial steady state is stable. If D = tr 2 4 det = (b R) 2 4cd R < 0 this is a stable spiral point. 41

42 The graphical Jacobian has the same signs: N a c ( ) α β γ 0 e d R a b tr(j) = α < 0 and det(j) = βγ > 0

43 Complex numbers Quadratic equation: aλ 2 + bλ + c = 0, with roots λ 1,2 = b ± b 2 4ac 2a = b ± D 2a where D = b 2 4ac What if D < 0? Define: i 2 = 1 or equivalently i = 1 Solve λ 2 = 3 by using i 2 = 1: λ 2 = i 2 3 or λ 1,2 = ±i 3 43

44 So if D < 0 write: λ 1,2 = b ± i D 2a Solve the equation λ 2 + 2λ + 10 = 0: λ 1,2 = 2 ± = 2 ± 36 2 In other words, λ 1 = 1 + 3i and λ 2 = 1 3i. = 2 ± 6i 2 A complex number z is written as z = α + iβ, where α is called the real part and iβ is called the imaginary part. These two solutions are complex conjugates: z 1 = a + ib and z 2 = a ib 44

45 Argand diagram: complex number as a vector: The complex plane imaginary part complex number z real part Addition of two complex numbers: adding their real parts, and add their imaginary parts. With z 1 = i and z 2 = 5 + 4i: z 1 + z 2 = (3 + 10i) + ( 5 + 4i) = i + 4i = i. 45

46 Multiplication works like (a + bx)(c + dx): z 1 z 2 = (3 + 10i)( 5 + 4i) = 3( 5) + 3 4i + 10i( 5) + 10i4i = i 50i + 40i 2 = 15 38i 40 = 55 38i. Note: (a + ib)(a ib) = a 2 + b 2 If z = a+ib, its modulus z = a 2 + b 2 (magnitude, length vector). Hence z z = z 2. (Used for division). 46

47 Mandelbrot set: z i = zi c, where c = a + bi is a point in the Argand diagram, and z 0 = 0. Black points remain bounded, colored points keep growing. The color indicates the number of iterations i = 1, 2,..., n required to reach a size of z n. Start with c = 0.5: 0.5, = 0.75, ,... 47

48 Linear ODEs { dx/dt = ax + by dy/dt = cx + dy λ with 1,2 = tr ± D 2 and { (a λi )x + by = 0 cx + (d λ i )y = 0 λ 1,2 = tr ± i D 2 or λ 1,2 = α ± iβ v 1 = k = k ( ) ( ) b b = k a λ 1 a (α + iβ) ( ) ( ) b 0 ik = kw a α β 1 ikw 2 where w 1 = ( b a α) and w2 = ( 0 β ) correspond to the real and imaginary parts of the eigenvector.

49 Similarly v 2 = k ( b a λ 2 ) = k ( b ) a (α iβ) = kw 1 + ikw 2 General solution: ( ) x(t) = C y(t) 1 (w 1 iw 2 )e (α+iβ)t + C 2 (w 1 + iw 1 )e (α iβ)t where the constants k are absorbed into C 1 and C 2. Euler s formula: hence e ix = cos x + i sin x or e ix = cos x i sin x e α+iβ = e α e iβ = e α (cos β + i sin β) 49

50 Hence from ( ) x(t) y(t) = C 1 (w 1 iw 2 )e (α+iβ)t + C 2 (w 1 + iw 1 )e (α iβ)t we obtain ( ) x(t) y(t) = C 1 (w 1 iw 2 )e αt (cos βt + i sin βt) + C 2 (w 1 + iw 2 )e αt (cos βt i sin βt) = e αt [C 1 (w 1 iw 2 )(cos βt + i sin βt) + C 2 (w 1 + iw 2 )(cos βt i sin βt)]. which dies out whenever α = tr/2 < 0. 50

51 Initial condition where t = 0, e αt = 1, cos βt = 1 and i sin βt = 0, ( ) x(0) = C y(0) 1 (w 1 iw 2 ) + C 2 (w 1 + iw 2 ) = w 1 (C 1 + C 2 ) + iw 2 (C 2 C 1 ), or x(0) = b(c 1 + C 2 ) and y(0) = (a α)(c 1 + C 2 ) + iβ(c 2 C 1 ) from which we solve the complex pair C 1 and C 2. Note that C 1 + C 2 should be real, whereas C 2 C 1 should be an imaginary number. 51

52 Lotka-Volterra model 1 1 (a) 0.05 (b) 0.05 (c) N dn/dt 0.5 hr, hn 0 0 x(t), y(t) hy hx 0 0 yx R R t 15 t t 15 t 20 dr dt = ar dn br2 crn, = drn en dt With a = b = c = d = 1, e = 0.5, R = 0.5 and N = 0.5, and h R = 0.05 and h N = 0 52

53 dr dt = ar br2 crn, dn dt = drn en, ( R, N) = with ( ) ( ) b R c R J = = d N. 0 be d ce d da eb c 0 ( e d da eb ), dc For a = b = c = d = 1 and e = 0.5, R = 0.5 and N = 0.5, and ( ) J = with D = implying that λ 1,2 = tr ± i D 2 or λ 1,2 = 0.5 ± i = 0.25 ± i Hence α = 0.25 and β = 0.43, the nontrivial state is stable, has a return time of 1/α = 4, and a wave length proportional to 1/β.

54 v 1 = ( ) i0.43) and v 2 = ( ) i0.43). ( x(t) y(t)) = e 0.25t [C 1 v 1 (cos 0.43t + i sin 0.43t) + C 2 v 2 (cos 0.43t i sin 0.43t)] x(t) = e 0.25t 0.5[(C 1 + C 2 ) cos 0.43t + (C 1 C 2 )i sin 0.43t] y(t) =... Using t = 0, e 0.25t = 1, cos 0.43t = 1, x(0) = 0.05, y(0) = 0, and sin 0.43t = 0: x(t) = e 0.25t [0.05 cos 0.433t sin 0.433t] y(t) = e 0.25t sin 0.433t 54

Matrices, Linearization, and the Jacobi matrix. y f x g y g J = dy/dt = g(x, y) Theoretical Biology, Utrecht University

Matrices, Linearization, and the Jacobi matrix. y f x g y g J = dy/dt = g(x, y) Theoretical Biology, Utrecht University Matrices, Linearization, and the Jacobi matrix { dx/dt f(x, y dy/dt g(x, y ( x f J y f x g y g λ 1, tr ± tr 4 det Theoretical Biology, Utrecht University i c Utrecht University, 018 Ebook publically available

More information

Matrices, Linearization, and the Jacobi matrix

Matrices, Linearization, and the Jacobi matrix Matrices, Linearization, and the Jacobi matrix { dx/dt f(x, y dy/dt g(x, y ( x f J y f x g y g λ 1, tr ± tr 4 det Alexander V. Panfilov, Kirsten H.W.J. ten Tusscher & Rob J. de Boer Theoretical Biology,

More information

Def. (a, b) is a critical point of the autonomous system. 1 Proper node (stable or unstable) 2 Improper node (stable or unstable)

Def. (a, b) is a critical point of the autonomous system. 1 Proper node (stable or unstable) 2 Improper node (stable or unstable) Types of critical points Def. (a, b) is a critical point of the autonomous system Math 216 Differential Equations Kenneth Harris kaharri@umich.edu Department of Mathematics University of Michigan November

More information

Calculus for the Life Sciences II Assignment 6 solutions. f(x, y) = 3π 3 cos 2x + 2 sin 3y

Calculus for the Life Sciences II Assignment 6 solutions. f(x, y) = 3π 3 cos 2x + 2 sin 3y Calculus for the Life Sciences II Assignment 6 solutions Find the tangent plane to the graph of the function at the point (0, π f(x, y = 3π 3 cos 2x + 2 sin 3y Solution: The tangent plane of f at a point

More information

Section 9.3 Phase Plane Portraits (for Planar Systems)

Section 9.3 Phase Plane Portraits (for Planar Systems) Section 9.3 Phase Plane Portraits (for Planar Systems) Key Terms: Equilibrium point of planer system yꞌ = Ay o Equilibrium solution Exponential solutions o Half-line solutions Unstable solution Stable

More information

7 Planar systems of linear ODE

7 Planar systems of linear ODE 7 Planar systems of linear ODE Here I restrict my attention to a very special class of autonomous ODE: linear ODE with constant coefficients This is arguably the only class of ODE for which explicit solution

More information

Eigenvalues. Matrices: Geometric Interpretation. Calculating Eigenvalues

Eigenvalues. Matrices: Geometric Interpretation. Calculating Eigenvalues Eigenvalues Matrices: Geometric Interpretation Start with a vector of length 2, for example, x =(1, 2). This among other things give the coordinates for a point on a plane. Take a 2 2 matrix, for example,

More information

Solutions to Dynamical Systems 2010 exam. Each question is worth 25 marks.

Solutions to Dynamical Systems 2010 exam. Each question is worth 25 marks. Solutions to Dynamical Systems exam Each question is worth marks [Unseen] Consider the following st order differential equation: dy dt Xy yy 4 a Find and classify all the fixed points of Hence draw the

More information

Lotka Volterra Predator-Prey Model with a Predating Scavenger

Lotka Volterra Predator-Prey Model with a Predating Scavenger Lotka Volterra Predator-Prey Model with a Predating Scavenger Monica Pescitelli Georgia College December 13, 2013 Abstract The classic Lotka Volterra equations are used to model the population dynamics

More information

Bifurcation Analysis of Non-linear Differential Equations

Bifurcation Analysis of Non-linear Differential Equations Bifurcation Analysis of Non-linear Differential Equations Caitlin McCann 0064570 Supervisor: Dr. Vasiev September 01 - May 013 Contents 1 Introduction 3 Definitions 4 3 Ordinary Differential Equations

More information

A = 3 B = A 1 1 matrix is the same as a number or scalar, 3 = [3].

A = 3 B = A 1 1 matrix is the same as a number or scalar, 3 = [3]. Appendix : A Very Brief Linear ALgebra Review Introduction Linear Algebra, also known as matrix theory, is an important element of all branches of mathematics Very often in this course we study the shapes

More information

Continuous time population models

Continuous time population models Continuous time population models Jaap van der Meer jaap.van.der.meer@nioz.nl Abstract Many simple theoretical population models in continuous time relate the rate of change of the size of two populations

More information

A VERY BRIEF LINEAR ALGEBRA REVIEW for MAP 5485 Introduction to Mathematical Biophysics Fall 2010

A VERY BRIEF LINEAR ALGEBRA REVIEW for MAP 5485 Introduction to Mathematical Biophysics Fall 2010 A VERY BRIEF LINEAR ALGEBRA REVIEW for MAP 5485 Introduction to Mathematical Biophysics Fall 00 Introduction Linear Algebra, also known as matrix theory, is an important element of all branches of mathematics

More information

Nonlinear differential equations - phase plane analysis

Nonlinear differential equations - phase plane analysis Nonlinear differential equations - phase plane analysis We consider the general first order differential equation for y(x Revision Q(x, y f(x, y dx P (x, y. ( Curves in the (x, y-plane which satisfy this

More information

MATH 215/255 Solutions to Additional Practice Problems April dy dt

MATH 215/255 Solutions to Additional Practice Problems April dy dt . For the nonlinear system MATH 5/55 Solutions to Additional Practice Problems April 08 dx dt = x( x y, dy dt = y(.5 y x, x 0, y 0, (a Show that if x(0 > 0 and y(0 = 0, then the solution (x(t, y(t of the

More information

Math 266: Phase Plane Portrait

Math 266: Phase Plane Portrait Math 266: Phase Plane Portrait Long Jin Purdue, Spring 2018 Review: Phase line for an autonomous equation For a single autonomous equation y = f (y) we used a phase line to illustrate the equilibrium solutions

More information

Complex Dynamic Systems: Qualitative vs Quantitative analysis

Complex Dynamic Systems: Qualitative vs Quantitative analysis Complex Dynamic Systems: Qualitative vs Quantitative analysis Complex Dynamic Systems Chiara Mocenni Department of Information Engineering and Mathematics University of Siena (mocenni@diism.unisi.it) Dynamic

More information

Constant coefficients systems

Constant coefficients systems 5.3. 2 2 Constant coefficients systems Section Objective(s): Diagonalizable systems. Real Distinct Eigenvalues. Complex Eigenvalues. Non-Diagonalizable systems. 5.3.. Diagonalizable Systems. Remark: We

More information

Systems of Algebraic Equations and Systems of Differential Equations

Systems of Algebraic Equations and Systems of Differential Equations Systems of Algebraic Equations and Systems of Differential Equations Topics: 2 by 2 systems of linear equations Matrix expression; Ax = b Solving 2 by 2 homogeneous systems Functions defined on matrices

More information

Section 8.2 : Homogeneous Linear Systems

Section 8.2 : Homogeneous Linear Systems Section 8.2 : Homogeneous Linear Systems Review: Eigenvalues and Eigenvectors Let A be an n n matrix with constant real components a ij. An eigenvector of A is a nonzero n 1 column vector v such that Av

More information

LS.2 Homogeneous Linear Systems with Constant Coefficients

LS.2 Homogeneous Linear Systems with Constant Coefficients LS2 Homogeneous Linear Systems with Constant Coefficients Using matrices to solve linear systems The naive way to solve a linear system of ODE s with constant coefficients is by eliminating variables,

More information

First Midterm Exam Name: Practice Problems September 19, x = ax + sin x.

First Midterm Exam Name: Practice Problems September 19, x = ax + sin x. Math 54 Treibergs First Midterm Exam Name: Practice Problems September 9, 24 Consider the family of differential equations for the parameter a: (a Sketch the phase line when a x ax + sin x (b Use the graphs

More information

MTH4101 CALCULUS II REVISION NOTES. 1. COMPLEX NUMBERS (Thomas Appendix 7 + lecture notes) ax 2 + bx + c = 0. x = b ± b 2 4ac 2a. i = 1.

MTH4101 CALCULUS II REVISION NOTES. 1. COMPLEX NUMBERS (Thomas Appendix 7 + lecture notes) ax 2 + bx + c = 0. x = b ± b 2 4ac 2a. i = 1. MTH4101 CALCULUS II REVISION NOTES 1. COMPLEX NUMBERS (Thomas Appendix 7 + lecture notes) 1.1 Introduction Types of numbers (natural, integers, rationals, reals) The need to solve quadratic equations:

More information

Systems Biology: Mathematics for Biologists. Kirsten ten Tusscher, Theoretical Biology, UU

Systems Biology: Mathematics for Biologists. Kirsten ten Tusscher, Theoretical Biology, UU Systems Biology: Mathematics for Biologists Kirsten ten Tusscher, Theoretical Biology, UU 1 Introductie: Wiskunde Maandag 20 febr en maandag 27 febr al 2 keer wiskunde les Onderwerpen nodig voor wiskunde

More information

Phase Plane Analysis

Phase Plane Analysis Phase Plane Analysis Phase plane analysis is one of the most important techniques for studying the behavior of nonlinear systems, since there is usually no analytical solution for a nonlinear system. Background

More information

Math 4B Notes. Written by Victoria Kala SH 6432u Office Hours: T 12:45 1:45pm Last updated 7/24/2016

Math 4B Notes. Written by Victoria Kala SH 6432u Office Hours: T 12:45 1:45pm Last updated 7/24/2016 Math 4B Notes Written by Victoria Kala vtkala@math.ucsb.edu SH 6432u Office Hours: T 2:45 :45pm Last updated 7/24/206 Classification of Differential Equations The order of a differential equation is the

More information

Copyright (c) 2006 Warren Weckesser

Copyright (c) 2006 Warren Weckesser 2.2. PLANAR LINEAR SYSTEMS 3 2.2. Planar Linear Systems We consider the linear system of two first order differential equations or equivalently, = ax + by (2.7) dy = cx + dy [ d x x = A x, where x =, and

More information

Math 312 Lecture Notes Linear Two-dimensional Systems of Differential Equations

Math 312 Lecture Notes Linear Two-dimensional Systems of Differential Equations Math 2 Lecture Notes Linear Two-dimensional Systems of Differential Equations Warren Weckesser Department of Mathematics Colgate University February 2005 In these notes, we consider the linear system of

More information

154 Chapter 9 Hints, Answers, and Solutions The particular trajectories are highlighted in the phase portraits below.

154 Chapter 9 Hints, Answers, and Solutions The particular trajectories are highlighted in the phase portraits below. 54 Chapter 9 Hints, Answers, and Solutions 9. The Phase Plane 9.. 4. The particular trajectories are highlighted in the phase portraits below... 3. 4. 9..5. Shown below is one possibility with x(t) and

More information

A matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and

A matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.

More information

Math Ordinary Differential Equations

Math Ordinary Differential Equations Math 411 - Ordinary Differential Equations Review Notes - 1 1 - Basic Theory A first order ordinary differential equation has the form x = f(t, x) (11) Here x = dx/dt Given an initial data x(t 0 ) = x

More information

A matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and

A matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.

More information

Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors Contents Eigenvalues and Eigenvectors. Basic Concepts. Applications of Eigenvalues and Eigenvectors 8.3 Repeated Eigenvalues and Symmetric Matrices 3.4 Numerical Determination of Eigenvalues and Eigenvectors

More information

Mathematics 1EM/1ES/1FM/1FS Notes, weeks 18-23

Mathematics 1EM/1ES/1FM/1FS Notes, weeks 18-23 2 MATRICES Mathematics EM/ES/FM/FS Notes, weeks 8-2 Carl Dettmann, version May 2, 22 2 Matrices 2 Basic concepts See: AJ Sadler, DWS Thorning, Understanding Pure Mathematics, pp 59ff In mathematics, a

More information

Ordinary Differential Equations (ODEs)

Ordinary Differential Equations (ODEs) Chapter 13 Ordinary Differential Equations (ODEs) We briefly review how to solve some of the most standard ODEs. 13.1 First Order Equations 13.1.1 Separable Equations A first-order ordinary differential

More information

Linearization of Differential Equation Models

Linearization of Differential Equation Models Linearization of Differential Equation Models 1 Motivation We cannot solve most nonlinear models, so we often instead try to get an overall feel for the way the model behaves: we sometimes talk about looking

More information

Physics: spring-mass system, planet motion, pendulum. Biology: ecology problem, neural conduction, epidemics

Physics: spring-mass system, planet motion, pendulum. Biology: ecology problem, neural conduction, epidemics Applications of nonlinear ODE systems: Physics: spring-mass system, planet motion, pendulum Chemistry: mixing problems, chemical reactions Biology: ecology problem, neural conduction, epidemics Economy:

More information

Find the general solution of the system y = Ay, where

Find the general solution of the system y = Ay, where Math Homework # March, 9..3. Find the general solution of the system y = Ay, where 5 Answer: The matrix A has characteristic polynomial p(λ = λ + 7λ + = λ + 3(λ +. Hence the eigenvalues are λ = 3and λ

More information

Basic Theory of Linear Differential Equations

Basic Theory of Linear Differential Equations Basic Theory of Linear Differential Equations Picard-Lindelöf Existence-Uniqueness Vector nth Order Theorem Second Order Linear Theorem Higher Order Linear Theorem Homogeneous Structure Recipe for Constant-Coefficient

More information

Old Math 330 Exams. David M. McClendon. Department of Mathematics Ferris State University

Old Math 330 Exams. David M. McClendon. Department of Mathematics Ferris State University Old Math 330 Exams David M. McClendon Department of Mathematics Ferris State University Last updated to include exams from Fall 07 Contents Contents General information about these exams 3 Exams from Fall

More information

Math 4381 / 6378 Symmetry Analysis

Math 4381 / 6378 Symmetry Analysis Math 438 / 6378 Smmetr Analsis Elementar ODE Review First Order Equations Ordinar differential equations of the form = F(x, ( are called first order ordinar differential equations. There are a variet of

More information

22.3. Repeated Eigenvalues and Symmetric Matrices. Introduction. Prerequisites. Learning Outcomes

22.3. Repeated Eigenvalues and Symmetric Matrices. Introduction. Prerequisites. Learning Outcomes Repeated Eigenvalues and Symmetric Matrices. Introduction In this Section we further develop the theory of eigenvalues and eigenvectors in two distinct directions. Firstly we look at matrices where one

More information

Department of Mathematics IIT Guwahati

Department of Mathematics IIT Guwahati Stability of Linear Systems in R 2 Department of Mathematics IIT Guwahati A system of first order differential equations is called autonomous if the system can be written in the form dx 1 dt = g 1(x 1,

More information

JUST THE MATHS UNIT NUMBER 9.8. MATRICES 8 (Characteristic properties) & (Similarity transformations) A.J.Hobson

JUST THE MATHS UNIT NUMBER 9.8. MATRICES 8 (Characteristic properties) & (Similarity transformations) A.J.Hobson JUST THE MATHS UNIT NUMBER 9.8 MATRICES 8 (Characteristic properties) & (Similarity transformations) by A.J.Hobson 9.8. Properties of eigenvalues and eigenvectors 9.8. Similar matrices 9.8.3 Exercises

More information

Math 215/255: Elementary Differential Equations I Harish N Dixit, Department of Mathematics, UBC

Math 215/255: Elementary Differential Equations I Harish N Dixit, Department of Mathematics, UBC Math 215/255: Elementary Differential Equations I Harish N Dixit, Department of Mathematics, UBC First Order Equations Linear Equations y + p(x)y = q(x) Write the equation in the standard form, Calculate

More information

Homogeneous Constant Matrix Systems, Part II

Homogeneous Constant Matrix Systems, Part II 4 Homogeneous Constant Matrix Systems, Part II Let us now expand our discussions begun in the previous chapter, and consider homogeneous constant matrix systems whose matrices either have complex eigenvalues

More information

Even-Numbered Homework Solutions

Even-Numbered Homework Solutions -6 Even-Numbered Homework Solutions Suppose that the matric B has λ = + 5i as an eigenvalue with eigenvector Y 0 = solution to dy = BY Using Euler s formula, we can write the complex-valued solution Y

More information

(V.C) Complex eigenvalues and other topics

(V.C) Complex eigenvalues and other topics V.C Complex eigenvalues and other topics matrix Let s take a closer look at the characteristic polynomial for a 2 2 namely A = a c f A λ = detλi A = λ aλ d bc b d = λ 2 a + dλ + ad bc = λ 2 traλ + det

More information

Nonlinear Autonomous Systems of Differential

Nonlinear Autonomous Systems of Differential Chapter 4 Nonlinear Autonomous Systems of Differential Equations 4.0 The Phase Plane: Linear Systems 4.0.1 Introduction Consider a system of the form x = A(x), (4.0.1) where A is independent of t. Such

More information

Homogeneous Liner Systems with Constant Coefficients

Homogeneous Liner Systems with Constant Coefficients Homogeneos Liner Systems with Constant Coefficients Jly, 06 The object of stdy in this section is where A is a d d constant matrix whose entries are real nmbers. As before, we will look to the exponential

More information

ENGI Duffing s Equation Page 4.65

ENGI Duffing s Equation Page 4.65 ENGI 940 4. - Duffing s Equation Page 4.65 4. Duffing s Equation Among the simplest models of damped non-linear forced oscillations of a mechanical or electrical system with a cubic stiffness term is Duffing

More information

Math 21b Final Exam Thursday, May 15, 2003 Solutions

Math 21b Final Exam Thursday, May 15, 2003 Solutions Math 2b Final Exam Thursday, May 5, 2003 Solutions. (20 points) True or False. No justification is necessary, simply circle T or F for each statement. T F (a) If W is a subspace of R n and x is not in

More information

Math 23: Differential Equations (Winter 2017) Midterm Exam Solutions

Math 23: Differential Equations (Winter 2017) Midterm Exam Solutions Math 3: Differential Equations (Winter 017) Midterm Exam Solutions 1. [0 points] or FALSE? You do not need to justify your answer. (a) [3 points] Critical points or equilibrium points for a first order

More information

ODE classification. February 7, Nasser M. Abbasi. compiled on Wednesday February 07, 2018 at 11:18 PM

ODE classification. February 7, Nasser M. Abbasi. compiled on Wednesday February 07, 2018 at 11:18 PM ODE classification Nasser M. Abbasi February 7, 2018 compiled on Wednesday February 07, 2018 at 11:18 PM 1 2 first order b(x)y + c(x)y = f(x) Integrating factor or separable (see detailed flow chart for

More information

Modelling and Mathematical Methods in Process and Chemical Engineering

Modelling and Mathematical Methods in Process and Chemical Engineering Modelling and Mathematical Methods in Process and Chemical Engineering Solution Series 3 1. Population dynamics: Gendercide The system admits two steady states The Jacobi matrix is ẋ = (1 p)xy k 1 x ẏ

More information

Differential Equations 2280 Sample Midterm Exam 3 with Solutions Exam Date: 24 April 2015 at 12:50pm

Differential Equations 2280 Sample Midterm Exam 3 with Solutions Exam Date: 24 April 2015 at 12:50pm Differential Equations 228 Sample Midterm Exam 3 with Solutions Exam Date: 24 April 25 at 2:5pm Instructions: This in-class exam is 5 minutes. No calculators, notes, tables or books. No answer check is

More information

With this expanded version of what we mean by a solution to an equation we can solve equations that previously had no solution.

With this expanded version of what we mean by a solution to an equation we can solve equations that previously had no solution. M 74 An introduction to Complex Numbers. 1 Solving equations Throughout the calculus sequence we have limited our discussion to real valued solutions to equations. We know the equation x 1 = 0 has distinct

More information

Repeated Eigenvalues and Symmetric Matrices

Repeated Eigenvalues and Symmetric Matrices Repeated Eigenvalues and Symmetric Matrices. Introduction In this Section we further develop the theory of eigenvalues and eigenvectors in two distinct directions. Firstly we look at matrices where one

More information

Calculus and Differential Equations II

Calculus and Differential Equations II MATH 250 B Second order autonomous linear systems We are mostly interested with 2 2 first order autonomous systems of the form { x = a x + b y y = c x + d y where x and y are functions of t and a, b, c,

More information

Further Mathematical Methods (Linear Algebra)

Further Mathematical Methods (Linear Algebra) Further Mathematical Methods (Linear Algebra) Solutions For The 2 Examination Question (a) For a non-empty subset W of V to be a subspace of V we require that for all vectors x y W and all scalars α R:

More information

MA 527 first midterm review problems Hopefully final version as of October 2nd

MA 527 first midterm review problems Hopefully final version as of October 2nd MA 57 first midterm review problems Hopefully final version as of October nd The first midterm will be on Wednesday, October 4th, from 8 to 9 pm, in MTHW 0. It will cover all the material from the classes

More information

Math 3301 Homework Set Points ( ) ( ) I ll leave it to you to verify that the eigenvalues and eigenvectors for this matrix are, ( ) ( ) ( ) ( )

Math 3301 Homework Set Points ( ) ( ) I ll leave it to you to verify that the eigenvalues and eigenvectors for this matrix are, ( ) ( ) ( ) ( ) #7. ( pts) I ll leave it to you to verify that the eigenvalues and eigenvectors for this matrix are, λ 5 λ 7 t t ce The general solution is then : 5 7 c c c x( 0) c c 9 9 c+ c c t 5t 7 e + e A sketch of

More information

Math 232, Final Test, 20 March 2007

Math 232, Final Test, 20 March 2007 Math 232, Final Test, 20 March 2007 Name: Instructions. Do any five of the first six questions, and any five of the last six questions. Please do your best, and show all appropriate details in your solutions.

More information

ECON : diff.eq. (systems).

ECON : diff.eq. (systems). ECON4140 018: diff.eq. (systems). FMEA 6.1 6.3 (Norw.: MA.1.3) You must know that if you are given two nonproportional solutions u 1 and u of a homogeneous linear nd order di.eq., you obtain the general

More information

MIDTERM REVIEW AND SAMPLE EXAM. Contents

MIDTERM REVIEW AND SAMPLE EXAM. Contents MIDTERM REVIEW AND SAMPLE EXAM Abstract These notes outline the material for the upcoming exam Note that the review is divided into the two main topics we have covered thus far, namely, ordinary differential

More information

ENGI Linear Approximation (2) Page Linear Approximation to a System of Non-Linear ODEs (2)

ENGI Linear Approximation (2) Page Linear Approximation to a System of Non-Linear ODEs (2) ENGI 940 4.06 - Linear Approximation () Page 4. 4.06 Linear Approximation to a System of Non-Linear ODEs () From sections 4.0 and 4.0, the non-linear system dx dy = x = P( x, y), = y = Q( x, y) () with

More information

Linear Algebra: Matrix Eigenvalue Problems

Linear Algebra: Matrix Eigenvalue Problems CHAPTER8 Linear Algebra: Matrix Eigenvalue Problems Chapter 8 p1 A matrix eigenvalue problem considers the vector equation (1) Ax = λx. 8.0 Linear Algebra: Matrix Eigenvalue Problems Here A is a given

More information

Some Notes on Linear Algebra

Some Notes on Linear Algebra Some Notes on Linear Algebra prepared for a first course in differential equations Thomas L Scofield Department of Mathematics and Statistics Calvin College 1998 1 The purpose of these notes is to present

More information

Solution: In standard form (i.e. y + P (t)y = Q(t)) we have y t y = cos(t)

Solution: In standard form (i.e. y + P (t)y = Q(t)) we have y t y = cos(t) Math 380 Practice Final Solutions This is longer than the actual exam, which will be 8 to 0 questions (some might be multiple choice). You are allowed up to two sheets of notes (both sides) and a calculator,

More information

MIT Final Exam Solutions, Spring 2017

MIT Final Exam Solutions, Spring 2017 MIT 8.6 Final Exam Solutions, Spring 7 Problem : For some real matrix A, the following vectors form a basis for its column space and null space: C(A) = span,, N(A) = span,,. (a) What is the size m n of

More information

Mathematical Economics. Lecture Notes (in extracts)

Mathematical Economics. Lecture Notes (in extracts) Prof. Dr. Frank Werner Faculty of Mathematics Institute of Mathematical Optimization (IMO) http://math.uni-magdeburg.de/ werner/math-ec-new.html Mathematical Economics Lecture Notes (in extracts) Winter

More information

Woods Hole Methods of Computational Neuroscience. Differential Equations and Linear Algebra. Lecture Notes

Woods Hole Methods of Computational Neuroscience. Differential Equations and Linear Algebra. Lecture Notes Woods Hole Methods of Computational Neuroscience Differential Equations and Linear Algebra Lecture Notes c 004, 005 William L. Kath MCN 005 ODE & Linear Algebra Notes 1. Classification of differential

More information

Applied Differential Equation. November 30, 2012

Applied Differential Equation. November 30, 2012 Applied Differential Equation November 3, Contents 5 System of First Order Linear Equations 5 Introduction and Review of matrices 5 Systems of Linear Algebraic Equations, Linear Independence, Eigenvalues,

More information

Math 215/255 Final Exam (Dec 2005)

Math 215/255 Final Exam (Dec 2005) Exam (Dec 2005) Last Student #: First name: Signature: Circle your section #: Burggraf=0, Peterson=02, Khadra=03, Burghelea=04, Li=05 I have read and understood the instructions below: Please sign: Instructions:.

More information

Lecture Notes 6: Dynamic Equations Part C: Linear Difference Equation Systems

Lecture Notes 6: Dynamic Equations Part C: Linear Difference Equation Systems University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 1 of 45 Lecture Notes 6: Dynamic Equations Part C: Linear Difference Equation Systems Peter J. Hammond latest revision 2017 September

More information

= A(λ, t)x. In this chapter we will focus on the case that the matrix A does not depend on time (so that the ODE is autonomous):

= A(λ, t)x. In this chapter we will focus on the case that the matrix A does not depend on time (so that the ODE is autonomous): Chapter 2 Linear autonomous ODEs 2 Linearity Linear ODEs form an important class of ODEs They are characterized by the fact that the vector field f : R m R p R R m is linear at constant value of the parameters

More information

Linear Algebra and ODEs review

Linear Algebra and ODEs review Linear Algebra and ODEs review Ania A Baetica September 9, 015 1 Linear Algebra 11 Eigenvalues and eigenvectors Consider the square matrix A R n n (v, λ are an (eigenvector, eigenvalue pair of matrix A

More information

MTH 464: Computational Linear Algebra

MTH 464: Computational Linear Algebra MTH 464: Computational Linear Algebra Lecture Outlines Exam 4 Material Prof. M. Beauregard Department of Mathematics & Statistics Stephen F. Austin State University April 15, 2018 Linear Algebra (MTH 464)

More information

We shall finally briefly discuss the generalization of the solution methods to a system of n first order differential equations.

We shall finally briefly discuss the generalization of the solution methods to a system of n first order differential equations. George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 Mathematical Annex 1 Ordinary Differential Equations In this mathematical annex, we define and analyze the solution of first and second order linear

More information

Symmetric and anti symmetric matrices

Symmetric and anti symmetric matrices Symmetric and anti symmetric matrices In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, matrix A is symmetric if. A = A Because equal matrices have equal

More information

STABILITY. Phase portraits and local stability

STABILITY. Phase portraits and local stability MAS271 Methods for differential equations Dr. R. Jain STABILITY Phase portraits and local stability We are interested in system of ordinary differential equations of the form ẋ = f(x, y), ẏ = g(x, y),

More information

Understand the existence and uniqueness theorems and what they tell you about solutions to initial value problems.

Understand the existence and uniqueness theorems and what they tell you about solutions to initial value problems. Review Outline To review for the final, look over the following outline and look at problems from the book and on the old exam s and exam reviews to find problems about each of the following topics.. Basics

More information

MATH 56A: STOCHASTIC PROCESSES CHAPTER 0

MATH 56A: STOCHASTIC PROCESSES CHAPTER 0 MATH 56A: STOCHASTIC PROCESSES CHAPTER 0 0. Chapter 0 I reviewed basic properties of linear differential equations in one variable. I still need to do the theory for several variables. 0.1. linear differential

More information

Systems of Linear ODEs

Systems of Linear ODEs P a g e 1 Systems of Linear ODEs Systems of ordinary differential equations can be solved in much the same way as discrete dynamical systems if the differential equations are linear. We will focus here

More information

1 Systems of Differential Equations

1 Systems of Differential Equations March, 20 7- Systems of Differential Equations Let U e an open suset of R n, I e an open interval in R and : I R n R n e a function from I R n to R n The equation ẋ = ft, x is called a first order ordinary

More information

Systems of differential equations Handout

Systems of differential equations Handout Systems of differential equations Handout Peyam Tabrizian Friday, November 8th, This handout is meant to give you a couple more example of all the techniques discussed in chapter 9, to counterbalance all

More information

Announcements Wednesday, November 01

Announcements Wednesday, November 01 Announcements Wednesday, November 01 WeBWorK 3.1, 3.2 are due today at 11:59pm. The quiz on Friday covers 3.1, 3.2. My office is Skiles 244. Rabinoffice hours are Monday, 1 3pm and Tuesday, 9 11am. Section

More information

Differential Equations and Modeling

Differential Equations and Modeling Differential Equations and Modeling Preliminary Lecture Notes Adolfo J. Rumbos c Draft date: March 22, 2018 March 22, 2018 2 Contents 1 Preface 5 2 Introduction to Modeling 7 2.1 Constructing Models.........................

More information

Eigenpairs and Diagonalizability Math 401, Spring 2010, Professor David Levermore

Eigenpairs and Diagonalizability Math 401, Spring 2010, Professor David Levermore Eigenpairs and Diagonalizability Math 40, Spring 200, Professor David Levermore Eigenpairs Let A be an n n matrix A number λ possibly complex even when A is real is an eigenvalue of A if there exists a

More information

APPM 2360: Final Exam 10:30am 1:00pm, May 6, 2015.

APPM 2360: Final Exam 10:30am 1:00pm, May 6, 2015. APPM 23: Final Exam :3am :pm, May, 25. ON THE FRONT OF YOUR BLUEBOOK write: ) your name, 2) your student ID number, 3) lecture section, 4) your instructor s name, and 5) a grading table for eight questions.

More information

6 EIGENVALUES AND EIGENVECTORS

6 EIGENVALUES AND EIGENVECTORS 6 EIGENVALUES AND EIGENVECTORS INTRODUCTION TO EIGENVALUES 61 Linear equations Ax = b come from steady state problems Eigenvalues have their greatest importance in dynamic problems The solution of du/dt

More information

Exam Basics. midterm. 1 There will be 9 questions. 2 The first 3 are on pre-midterm material. 3 The next 1 is a mix of old and new material.

Exam Basics. midterm. 1 There will be 9 questions. 2 The first 3 are on pre-midterm material. 3 The next 1 is a mix of old and new material. Exam Basics 1 There will be 9 questions. 2 The first 3 are on pre-midterm material. 3 The next 1 is a mix of old and new material. 4 The last 5 questions will be on new material since the midterm. 5 60

More information

Worksheet # 2: Higher Order Linear ODEs (SOLUTIONS)

Worksheet # 2: Higher Order Linear ODEs (SOLUTIONS) Name: November 8, 011 Worksheet # : Higher Order Linear ODEs (SOLUTIONS) 1. A set of n-functions f 1, f,..., f n are linearly independent on an interval I if the only way that c 1 f 1 (t) + c f (t) +...

More information

Mathematical Modeling I

Mathematical Modeling I Mathematical Modeling I Dr. Zachariah Sinkala Department of Mathematical Sciences Middle Tennessee State University Murfreesboro Tennessee 37132, USA November 5, 2011 1d systems To understand more complex

More information

Systems of Linear Differential Equations Chapter 7

Systems of Linear Differential Equations Chapter 7 Systems of Linear Differential Equations Chapter 7 Doreen De Leon Department of Mathematics, California State University, Fresno June 22, 25 Motivating Examples: Applications of Systems of First Order

More information

Differential Equations Revision Notes

Differential Equations Revision Notes Differential Equations Revision Notes Brendan Arnold January 18, 2004 Abstract These quick refresher notes will probably only be useful for those with a univsersity level maths. They were written primarily

More information

Matrices and Deformation

Matrices and Deformation ES 111 Mathematical Methods in the Earth Sciences Matrices and Deformation Lecture Outline 13 - Thurs 9th Nov 2017 Strain Ellipse and Eigenvectors One way of thinking about a matrix is that it operates

More information

PH.D. PRELIMINARY EXAMINATION MATHEMATICS

PH.D. PRELIMINARY EXAMINATION MATHEMATICS UNIVERSITY OF CALIFORNIA, BERKELEY SPRING SEMESTER 207 Dept. of Civil and Environmental Engineering Structural Engineering, Mechanics and Materials NAME PH.D. PRELIMINARY EXAMINATION MATHEMATICS Problem

More information

Lecture 38. Almost Linear Systems

Lecture 38. Almost Linear Systems Math 245 - Mathematics of Physics and Engineering I Lecture 38. Almost Linear Systems April 20, 2012 Konstantin Zuev (USC) Math 245, Lecture 38 April 20, 2012 1 / 11 Agenda Stability Properties of Linear

More information

Kinematics of fluid motion

Kinematics of fluid motion Chapter 4 Kinematics of fluid motion 4.1 Elementary flow patterns Recall the discussion of flow patterns in Chapter 1. The equations for particle paths in a three-dimensional, steady fluid flow are dx

More information