MATH 213 Linear Algebra and Ordinary Differential Equations Spring 2015 Study Sheet for Final Exam. Topics

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1 MATH 213 Linear Algebra and Ordinary Differential Equations Spring 2015 Study Sheet for Final Exam This study sheet will not be allowed during the test. Books and notes will not be allowed during the test. Calculators and cell phones will not be allowed during the test. 1. Dot product 2. The complex numbers 3. Vector spaces 4. Eigenvalues, eigenvectors, eigenspaces Topics 5. General solutions and particular solutions of differential equations 6. Separable differential equations 7. Applications of separable differential equations (for example, exponential growth, radioactive decay, Newton s Law of Cooling) 8. First order linear differential equations 9. Higher order linear differential equations 10. Homogeneous higher order linear differential equations 11. Homogeneous higher order linear differential equations with constant coefficients 12. Undetermined coefficients 13. Springs 14. Systems of first order linear differential equations 15. Systems of first order linear differential equations: homogeneous equations with constant coefficients via eigenvalues 1

2 Practice Problems from Edwards & Penney, Diff. Eq. & Lin. Alg., 3rd ed. Section 4.6: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 30, 31 Complex Numbers Handout: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46 Section 4.7: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 Section 6.1: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 33, 34, 36 Section 1.1: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 32, 33, 34, 35, 36 Section 1.4: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 33, 34, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49 Section 1.5: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 33, 34, 35, 36, 37 Section 5.3: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26 Section 5.4: 15, 16, 17, 18, 19, 20, 21 Section 5.5: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 31, 32, 33, 34, 35, 36, 37, 38, 39, 40 Section 7.3: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 2

3 Some Important Concepts and Formulas 1. Dot Product Let n N. Let x = defined by [ x1 ] x 2 and y =. x n [ y1 ] y 2 be vectors in R n. The dot product of x and y is. y n x y = x 1 y 1 + x 2 y x n y n. 2. Properties of the Dot Product Let n N. Let x,y,z R n, and let c R. 1. x y = y x (Symmetry Law). 2. x (y + z) = x y + x z (Distributive Law). 3. (cx) y = c(x y) (Homogeneity Law). 4. x x 0, and x x = 0 if and only if x = 0 (Positive Definite Law). 5. x x = x x = 0 = x x y = 1 2 ( x 2 + y 2 x y 2 ) (Polarization Identity). 8. x y x y (Cauchy-Schwarz Inequality). 9. x + y x + y (Triangle Inequality). 3. Geometry of the Dot Product Let n N. Let x,y R n. 1. Let θ be the angle between x and y. Then x y = x y cosθ. 2. The vectors x and y are orthogonal if x y = Let v 1,...,v k R n be vectors. The vectors v 1,...,v k are orthogonal if v i v j = 0 for all values of i and j such that i j. 4. The vectors v 1,...,v k are orthonormal if they are orthogonal and if v i = 1 for all i. 3

4 4. The Complex Numbers 1. The number i is the number such that i 2 = A complex number is any number of the form a + bi, where a,b R. 3. The set of all complex numbers is denoted C. 4. Let z = a + bi. The real part of z is a, and the imaginary part of z is b. 5. The modulus of z, denoted z, is defined by z = a 2 + b The complex conjugate of z, denoted z, is defined by z = a bi. 5. The Complex Numbers: Addition, and Multiplication Let z = a + bi and w = c + di, and let t R. 1. z + w = (a + c) + (b + d)i. 2. z w = (a c) + (b d)i. 3. zw = (ab bd) + (ad + bc)i. 4. z w = a+bi c+di = (a+bi)(c di) (c+di)(c di) = (ab+bd)+(bc ad)i c 2 +d tz = ta +tbi. 6. The Complex Numbers: Polar Form Let z = a + bi. Suppose that (r,θ) are the polar coordinates of the point (a,b) in R a = r cosθ and b = r sinθ. 2. r = z = a 2 + b 2 and tanθ = b a. 3. The argument of z is the angle θ. 4. The polar form of z is z = r(cosθ + isinθ). 4

5 7. The Complex Numbers: Multiplication and Division in Polar Form Let z = r(cosθ + isinθ) and w = s(cosφ + isinφ), and let n N zw = rs(cos(θ + φ) + isin(θ + φ)). z w = r (cos(θ φ) + isin(θ φ)). s z n = r n (cosnθ + isinnθ). 4. The number z has n distinct n th roots, which are given by ( ) ( ) w k = r 1 θ + 2kπ θ + 2kπ n (cos + i n n for k {0,1,...,n 1}. 8. Euler s Formula Let a + bi C. e a+bi = e a (cosb + isinb). 9. Vector Spaces A vector space is a set V together with an operation called addition, and an operation called multiplication by scalars (which are real numbers), which satisfy the following eight properties. Let u,v,w V, and let s,t R. 1. u + v = v + u (Commutative Law). 2. u + (v + w) = (u + v) + w (Associative Law). 3. u + 0 = u and u + 0 = A (Identity Law). 4. u + ( u) = 0 and ( u) + u = 0 (Inverses Law). 5. s(u + v) = su + sv (Distributive Law). 6. (s +t)u = su +tu (Distributive Law). 7. s(tu) = (st)u. 8. 1u = u. 5

6 10. Eigenvalues and Eigenvectors Let n N. Let A M n n (R). 1. Let λ R. The number λ is an eigenvalue of A if there is a non-zero vector v R n such that Av = λv; such a vector v is an eigenvector of A corresponding to λ. 2. The characteristic polynomial of A is the polynomial det(a λi n ). 3. The eigenvalues of A, if there are any, are the roots of the characteristic polynomial. For each eigenvalue, the corresponding eigenvectors can be found by substituting the eigenvalue λ into the equation (A λi n )v = 0, and finding non-zero solutions for v. 11. Eigenspaces Let n N. Let A M n n (R). Let λ be an eigenvalue of A. 1. The eigenspace of A associated with λ is the solution space of the equation (A λi n )v = The eigenspace of A associated with λ is a subspace of R n. 3. Let λ 1,...,λ k R be distinct real eigenvalues of A, and let v 1,...,v k R n be eigenvectors corresponding to these eigenvalues. Then v 1,...,v k are linearly independent. 4. Suppose that A has n distinct eigenvalues. Then there is a basis for R n consisting of eigenvectors of A. 12. Ordinary Differential Equations 1. An ordinary differential equation, abbreviated ODE, is an equation that involves the derivatives of single variable functions, all of which have the same variable. 2. A partial differential equation, abbreviated PDE, is an equation that involves partial derivatives of multi-variable functions. 3. A differential equation has order n if the highest derivative in the differential equation is an n th derivative. 6

7 13. General Solutions, Particular Solutions and Initial Values 1. A particular solution of an ODE is a solution that does not involve any constants. 2. A general solution of an ODE is a solution with constants. 3. The general solution of an n th order ODE will usually have n constants. 4. A singular solution of an ODE is a particular solution that cannot be obtained from the general solution by using any possible numerical values of the constants in the general solution. 5. Initial conditions of an ODE are a collections of conditions given by values of the function and/or its derivatives at specific inputs. 6. An initial value problem is an ODE together with initial conditions. 14. Separable Ordinary Differential Equations 1. A separable ODE is an ODE that can be brought into the form dy dx = g(x)h(y), for some functions g(x) and h(y). 2. A separable ODE can be solved by bringing all instances of one variable to the left side of the equality, and all instances of the other variable to the right side of the equality, and then integrating both sides. 15. Applications of Separable Differential Equations Exponential Growth of a Population Let P(t) be a population at time t. where k > 0. dp dt = kp, Radioactive Decay Let x(t) be the amount of a radioactive material at time t. where k < 0. dx dt = kx, 7

8 Newton s Law of Cooling Let T (t) be the temperature of an object at time t, and let A be the ambient temperature. where k > 0. dt dt = k(t A), 16. First Order Linear Differential Equations 1. A first order linear differential equation is an ODE of the form for some functions P(x) and Q(x). dy + P(x)y = Q(x), dx 2. To solve a first order linear ODE, multiply it by ρ(x) = e P(x)dx, and obtain d [yρ(x)] = Q(x)ρ(x). dx 17. Higher Order Linear Differential Equations 1. A linear differential equation of order n is an ODE of the form a 0 (x)y (n) + a 1 (x)y (n 1) + a 2 (x)y (n 2) + + a n 1 (x)y + a n (x)y = g(x), for some functions a 0 (x),a 1 (x),...,a n (x) and g(x). 2. Let y p be a particular solution of this ODE; that is, the solution y p has no constants in it. Let y c be the general solution of the associated homogeneous linear ODE, which is a 0 (x)y (n) + a 1 (x)y (n 1) + a 2 (x)y (n 2) + + a n 1 (x)y + a n (x)y = 0. Then y = y c +y p is the general solution of the original non-homogeneous linear ODE. 8

9 18. Homogeneous Higher Order Linear Differential Equations 1. A homogeneous linear differential equation of order n is an ODE of the form a 0 (x)y (n) + a 1 (x)y (n 1) + a 2 (x)y (n 2) + + a n 1 (x)y + a n (x)y = 0. for some functions a 0 (x),a 1 (x),...,a n (x). 2. The solution set of this ODE is a subspace of the set of all functions R R. That is, if y 1,...,y k are solutions, and if c 1,...,c k R, then y = c 1 y c k y k is a solution. 19. Homogeneous Higher Order Linear Differential Equations with Constant Coefficients 1. A homogeneous linear differential equation with constant coefficients of order n is an ODE of the form for some a 0,a 1,...,a n R. a 0 y (n) + a 1 y (n 1) + a 2 y (n 2) + a n 1 y + a n y = 0, 2. The characteristic equation of this ODE is a 0 r n + a 1 r n 1 + a 2 r n 2 + a n 1 r + a n = To solve the ODE, solve the characteristic equation for all its roots. Some of the roots will be real, and some will be complex (which come in pairs of the form a ± bi). Each root has a multiplicity, which can be found by factoring the characteristic equation. For each root, obtain the following solutions: root multiplicity solution r 1 e rx r k e rx,xe rx,x 2 e rx,...,x k 1 e rx a ± bi 1 e ax cos(bx),e ax sin(bx) a ± bi k e ax cos(bx),e ax sin(bx),...,x k 1 e ax cos(bx),x k 1 e ax sin(bx). 4. If y 1,...,y n are the solutions obtained in this way, then the general solution of the ODE is y = c 1 y c n y n, where c 1,...,c n are constants. 9

10 20. Undetermined Coefficients 1. A linear differential equation with constant coefficients of order n is an ODE of the form a 0 y (n) + a 1 y (n 1) + a 2 y (n 2) + a n 1 y + a n y = f (x), for some a 0,a 1,...,a n R, and some function f (x). 2. To find a particular solution, here are some common types of functions f (x), and what to guess for particular solutions: function p m (x) ae rx p m (x)e rx asinkx acoskx ae rx sinkx ae rx coskx p m (x)sinkx p m (x)coskx guess A 0 + A 1 x + A m x m Ae rx (A 0 + A 1 x + A m x m )e rx Asinkx + Bcoskx Asinkx + Bcoskx e rx (Asinkx + Bcoskx) e rx (Asinkx + Bcoskx) (A 0 + A 1 x + A m x m )sinkx + (B 0 + B 1 x + B m x m )coskx (A 0 + A 1 x + A m x m )sinkx + (B 0 + B 1 x + B m x m )coskx 3. To solve the ODE, first find the general solution y c of the associated homogeneous ODE. Then make a guess y p for a particular solution of the original ODE, but if any part of the guess overlaps with y c, multiply that part of the guess by whatever power of x is needed to avoid any overlap. Solve for the constants in y p. Then y = y c + y p is the general solution of the original non-homogeneous ODE. 21. Springs The ODE for the motion of a spring is mx + cx + kx = F(t), where m is the mass, and c is the resistance, and k is the spring constant, and F(t) is a forcing function. 10

11 22. Springs: Free Undamped Motion The spring has free undamped motion when c = 0 and F(t) = 0. The ODE is then Let ω 0 = k m. mx + kx = The general solution is x(t) = Acosω 0 t + Bsinω 0 t. 2. Let C = A 2 + B 2, and let α be such that cosα = A C and sinα = B C. 3. The general solution is x(t) = C cos(ω 0 t α). 23. Springs: Free Damped Motion The spring has free undamped motion when F(t) = 0. The ODE is then Let ω 0 = k m and p = c 2m. mx + cx + kx = 0. Underdamped Case Suppose that c 2 < 4km. 1. Let ω 1 = ω0 2 p2 = 4km c 2 2m. 2. The general solution is x(t) = e pt (Acosω 1 t + Bsinω 1 t). 3. Let C = A 2 + B 2, and let α be such that cosα = A C and sinα = B C. 4. The general solution is x(t) = Ce pt cos(ω 1 t α). Overdamped Case Suppose that c 2 > 4km. 1. Let r 1 = p + p 2 ω0 2 and r 2 = p p 2 ω The general solution is x(t) = Ae r 1t + Be r 2t. Critically Damped Case Suppose that c 2 = 4km. 1. The general solution is x(t) = (A + Bx)e pt. 11

12 24. Springs: Undamped Forced Oscillation The spring has undamped forced oscillation when c = 0 and F(t) 0. Consider the case when F(t) = F 0 cosωt. The ODE is then Let ω 0 = k m. mx + kx = F 0 cosωt. Beats Suppose that ω ω 0. Resonance Suppose that ω = ω Systems of First Order Linear Differential Equations 1. A system of first order linear differential equations is a system of n ODEs of the form x 1(t) = p 11 (t)x 1 (t) + p 12 (t)x 2 (t) + + p 1n (t)x n (t) + f 1 (t) x 2(t) = p 21 (t)x 1 (t) + p 22 (t)x 2 (t) + + p 2n (t)x n (t) + f 2 (t). x n(t) = p n1 (t)x 1 (t) + p n2 (t)x 2 (t) + + p nn (t)x n (t) + f n (t), for some functions p 11 (t), p 12 (t),..., p nn (t) and f 1 (t), f 2 (t),..., f n (t). 2. The system of ODEs is homogeneous if all the functions f 1 (t), f 2 (t),..., f n (t) are zero; otherwise, the system of ODEs is non-homogeneous. 12

13 26. Systems of First Order Linear Differential Equations: Matrices A system of first order linear differential equations can be written in the form x 1(t) = p 11 (t)x 1 (t) + p 12 (t)x 2 (t) + + p 1n (t)x n (t) + f 1 (t) x 2(t) = p 21 (t)x 1 (t) + p 22 (t)x 2 (t) + + p 2n (t)x n (t) + f 2 (t). x n(t) = p n1 (t)x 1 (t) + p n2 (t)x 2 (t) + + p nn (t)x n (t) + f n (t), for some functions p 11 (t), p 12 (t),..., p nn (t) and f 1 (t), f 2 (t),..., f n (t). Define the matrices (with entries that are functions) p 11 (t) p 12 (t) p 13 (t) p 1n (t) p 21 (t) p 22 (t) p 23 (t) p 2n (t) P(t) =..... p n1 (t) p n2 (t) p n3 (t) p nn (t) and x 1 (t) x 2 (t) x(t) =. x n (t) f 1 (t) f 2 (t) and f (t) =.. f n (t) Then the system of system of first order linear differential equations is equivalent to the matrix equation x (t) = P(t)x(t) + f (t). 13

14 27. Systems of First Order Linear Differential Equations: Homogeneous Equations with Constant Coefficients A system of first order homogeneous linear ODEs with constant coefficients can be written in the form x 1(t) = a 11 x 1 (t) + a 12 x 2 (t) + + a 1n x n (t) x 2(t) = a 21 x 1 (t) + a 22 x 2 (t) + + a 2n x n (t). x n(t) = a n1 x 1 (t) + a n2 x 2 (t) + + a nn x n (t), for some real numbers a 11,a 12,...,a nn. Define the matrices a 11 a 12 a 1n a 21 a 22 a 2n A =.... a m1 a m2 a mn x 1 (t) x 2 (t) and x(t) =.. x n (t) Then the system of system of first order linear differential equations is equivalent to the matrix equation x (t) = Ax(t). For convenience it is possible to write this matrix equation as x = Ax, with the understanding that x is a function of t. 28. Systems of First Order Linear Differential Equations: Homogeneous Equations with Constant Coefficients via Eigenvalues Let x = Ax be a system of homogeneous first order linear ODEs with constant coefficients. 1. If r is an eigenvalue of A with corresponding eigenvector v, then x(t) = ve rt is a solution of the system of ODEs x = Ax. 2. If the matrix A is n n, and if A has n distinct real eigenvalues, then there are n linearly independent solutions, which together yield the general solution of the system of ODEs. 14