Higher Order ODE's, (3A)

Higher Order ODE's, (3A) Initial Value Problems, and Boundary Value Problems

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The Properties of a Line Equation (1) y = x f (1) = 1 f (2) = f (1+1)=f (1) + f (1) = 2 f (2) = 2 f (3) = f (2+1)=f (2) + f (1) = 3 f (3) = 3 f (4) = f (3+1)=f (3) + f (1) = 4 multiples of a unit y = x f (0.1) = 0.1 f (0.1) = f (0.1 1)=0.1 f (1) = 0.1 f (0.5) = 0.5 f (0.5) = f (0.5 1)=0.5 f (1) = 0.5 f (1.5) = 1.5 f (1.5) = f (1.5 1)=1.5 f (1) = 1.5 fractions of a unit Higher Order ODEs (3A) 3

The Properties of a Line Equation (2) multiples of a unit y = x fractions of a unit f (1) = 1 f (2) = 2 f (3) = 3 f (4) = 4 f (0.5) = 0.5 f (1.0) = 1.0 f (1.5) = 1.5 f (2.0) = 2.0 f (0.1) = 0.1 f (0.2) = 0.2 f (0.3) = 0.3 f (0.4) = 0.4 y = x f (x 1 + x 2 ) = f (x 1 ) + f ( x 2 ) f (k x) = k f (x) Higher Order ODEs (3A) 4

Linearity Property a( x 1 + x 2 ) a x 2 a x 1 y = a x Additivity f (x 1 + x 2 ) = f (x 1 ) + f ( x 2 ) a (x 1 + x 2 ) = a (x 1 ) + a ( x 2 ) x 1 x 2 x 1 +x 2 y = a x k a x a x Homogeneity f (k x) = k f ( x) x k x a (k x) = k(a x) Higher Order ODEs (3A) 5

Linearity & Affinity Linearity Affinity y = a x y = a x + b Translation a x 1 + b a x 2 + b a(x 1 + x 2 ) + b a x 1 + b a k x 1 + b k (a x + b) from linear + -ity. from the Latin, affinis, "connected with" f (x 1 + x 2 ) = f (x 1 ) + f ( x 2 ) f (k x) = k f (x) Additivity Homogeneity Additivity Homogeneity Higher Order ODEs (3A) 6

Linear Map Additivity f (x 1 + x 2 ) = f (x 1 ) + f (x 2 ) x 1 f (x) f (x 1 ) x 1 + x 2 f (x) f (x 1 ) + f ( x 2 ) x 2 f (x) f (x 2 ) Homogeneity f (k f (k x) = k f (x) f ( x) x f (x) f (x) k x f (x) k f ( x) Higher Order ODEs (3A) 7

Linear Operators Additivity d d f [f + g] = d x d x + d g d x f (x) d d x f ' (x) f (x) + g( x) d d x f ' (x ) + g' (x) g( x) d g' (x) d x Homogeneity d d x [k f ] = k d f d x f (k x) = k f ( x) f (x) d f ' (x) k f ( x) d k f '( x) d x d x Higher Order ODEs (3A) 8

Linear Systems Additivity S{g 1 (x) + g 2 (x)} = S {g 1 ( x)} + S{g 2 ( x)} g 1 (x) S y 1 ( x) g 1 (x) + g 2 ( x) S y 1 ( x) + y 2 ( x) g 2 ( x) S y 2 ( x) d 2 y d x 2 + a 1 d y d x + a 2 y = g( x) Homogeneity f (k x) = k f ( x) S{k f (x)} = k S{f (x)} g( x) S y(x) k g( x) S k y(x) Higher Order ODEs (3A) 9

Differential Operator Differential Operator D y(x) y '( x) D = d dx D y = d y dx D( y) = d y dx D( y(x)) = d y dx N-th Order Differential Operator y(x) L L( y(x)) L = a n ( x) D n + a n 1 ( x) D n 1 + + a 1 ( x) D + a 0 ( x) L( y) = {a n (x) D n + a n 1 (x) D n 1 + + a 1 ( x) D + a 0 (x)}( y) L( y) = a n ( x) D n ( y) + a n 1 ( x) D n 1 ( y) + + a 1 ( x)d( y) + a 0 ( x)( y) L( y) = a n (x) dn y d x n + a n 1 (x) dn 1 y d x n 1 + + a 1 ( x) d y d x + a 0( x) y Higher Order ODEs (3A) 10

Examples f (x) d d x f ' (x) d f d x = f '(x) D (D f ) = d dx ( d f d x ) = f ' ' (x) f (x) D f ' (x) D f = f '( x) D 2 f = d2 f d x 2 = f ' ' (x) Differential Operator : Linear D (c f (x)) = c D f (x) D (f (x) + g(x)) = D f ( x) + D g(x) D (α f ( x) + β g(x)) = α D f ( x) + β D g(x) n-th order Differential Operator L = a n (x)d n + a n 1 (x) D n 1 + + a 1 ( x)d + a 0 (x) (D 2 + 2 D + 1) f (x) = D 2 f ( x) + 2 D f ( x) + f (x) = f ' '( x) + 2 f '(x) + f ( x) n-th order Differential Equations using the Differential Operator y' ' + 5 y '+2 y = 3 x D 2 + 5 D + 2 = L L( y) = 0 L( y) = g(x) y' ' + 5 y '+2 y = 0 y' ' + 5 y '+2 y = 3 x Higher Order ODEs (3A) 11

Linear System Linear System g( x) S y (x) a n (x) dn y d x n + a n 1( x) dn 1 y d x n 1 + + a 1( x) d y d x + a 0( x) y = g(x) (a n (x)d n + a n 1 (x)d n 1 + + a 1 ( x) D + a 0 (x))( y( x)) = g( x) L( y(x)) = g( x) Higher Order ODEs (3A) 12

Linear Differential Equations a n (x) dn y d x n + a n 1( x) dn 1 y d x n 1 + + a 1( x) d y d x + a 0( x) y = g(x) a n (x) dn y d x n + a n 1( x) dn 1 y d x n 1 + + a 1( x) d y d x + a 0( x) y = 0 Non-homogeneous Equation Homogeneous Equation L = a n ( x) D n + a n 1 ( x) D n 1 + + a 1 ( x) D + a 0 ( x) y(x) L L( y(x)) L( y) = a n ( x) dn y d x n + a n 1(x) dn 1 y d x n 1 + + a 1(x) d y d x + a 0( x) y Higher Order ODEs (3A) 13

Linear Differential Equations Linear Equation - Additivity a n (x) dn y 1 d x n a n (x) dn y 2 d x n + a n 1(x) d n 1 y 1 d x n 1 + + a 1(x) d y 1 d x + a 0(x) y 1 = L( y 1 ) y 1 (x) L(y 1 (x)) + a n 1(x) d n 1 y 2 d x + + a 1(x) d y 2 n 1 d x + a 0(x) y 2 = L( y 2 ) y 2 (x) L(y 2 (x)) a n ( x) dn ( y 1 + y 2 ) d x n + a n 1 ( x) d n 1 ( y 1 + y 2 ) + + a d x n 1 1 ( x) d( y 1+ y 2 ) d x + a 0 (x)( y 1 + y 2 ) = L( y 1 + y 2 ) y 1 + y 2 L(y 1 )+L( y 2 ) Linear Equation - Homogeneity a n ( x) dn y 1 d x n + a n 1(x) d n 1 y 1 d x + + a 1(x) d y 1 n 1 d x + a 0(x) y 1 = L( y 1 ) y 1 (x) L( y 1 (x)) a n (x) dn k y 1 d x n + a n 1 (x) d n 1 k y 1 + + a d x n 1 1 (x) d k y 1 d x + a 0 (x)k y 1 = L(k y) k y 1 ( x) k L( y 1 (x)) Higher Order ODEs (3A) 14

Linear Differential Equation Solutions Linear Differential Equation Solution Additivity a n ( x) dn y 1 d x n + a n 1(x) d n 1 y 1 d x + + a 1(x) d y 1 n 1 d x + a 0(x) y 1 = g 1 (x) g 1 (x) y 1 (x) a n ( x) dn y 2 d x n + a n 1(x) d n 1 y 2 d x + + a 1(x) d y 2 n 1 d x + a 0(x) y 2 = g 2 (x) g 2 (x) y 2 (x) a n ( x) dn ( y 1 + y 2 ) d x n + a n 1 ( x) d n 1 ( y 1 + y 2 ) + + a d x n 1 1 ( x) d( y 1+ y 2 ) d x + a 0 (x)( y 1 + y 2 ) = g 1 (x) + g 2 (x) Superposition g 1 +g 2 y 1 + y 2 Linear Differential Equation Solution Homogeneity a n (x) dn y 1 d x n + a n 1(x) d n 1 y 1 d x + + a 1(x) d y 1 n 1 d x + a 0(x) y 1 = g 1 (x) g 1 (x) y 1 (x) a n (x) dn k y 1 d x n + a n 1 ( x) d n 1 k y 1 + + a d x n 1 1 ( x) d k y 1 d x + a 0 ( x) k y 1 = k g 1 ( x) k g 1 (x) k y 2 ( x) Higher Order ODEs (3A) 15

Homogeneous Equation a n (x) dn y d x n + a n 1( x) dn 1 y d x n 1 + + a 1( x) d y d x + a 0( x) y = g(x) a n (x) dn y d x n + a n 1( x) dn 1 y d x n 1 + + a 1( x) d y d x + a 0( x) y = 0 Associated Homogeneous Equation d n y a n d x + a d n 1 y n n 1 d x + + a d y n 1 1 d x + a 0 y = g( x) d n y a n d x + a d n 1 y n n 1 d x + + a d y n 1 1 d x + a 0 y = 0 Associated Homogeneous Equation with constant coefficients a n m n + a n 1 m n 1 + + a 1 m + a 0 = 0 Auxiliary Equation m = m 1, m 2,, m n y = c 1 e m 1 x + c 2 e m 2x + + c n e m n x n solutions of the Auxiliary Equation General Solutions of the Homogeneous Equation Higher Order ODEs (3A) 16

Non-homogeneous Equation a n (x) dn y d x n + a n 1( x) dn 1 y d x n 1 + + a 1( x) d y d x + a 0( x) y = g(x) a n (x) dn y d x n + a n 1( x) dn 1 y d x n 1 + + a 1( x) d y d x + a 0( x) y = 0 Associated Homogeneous Equation d n y a n d x + a d n 1 y n n 1 d x + + a d y n 1 1 d x + a 0 y = g( x) d n y a n d x + a d n 1 y n n 1 d x + + a d y n 1 1 d x + a 0 y = 0 Associated Homogeneous Equation with constant coefficients Particular Solution y = c 1 e m 1 x + c 2 e m 2 x + + c n e m n x + y p ( x) General Solutions of the Non-homogeneous Equation Higher Order ODEs (3A) 17

Initial Value Problem d n y d x n = f (x, y, y ',, y (n 1) ) on some interval I containing x 0 General Form d n 1 d x n 1 y(x 0) = k n 1 d d x y(x 0) = y' (x 0 ) = k 1 n Initial Conditions at x = x 0 y(x 0 ) = y(x 0 ) = k 0 IVP Higher Order ODEs (3A) 18

Initial Value Problem variable coefficients a n ( x) dn y d x n + a n 1(x) dn 1 y d x n 1 + + a 1(x) d y d x + a 0(x) y = g(x) Linear Equation with variable coefficients d n 1 d x n 1 y(x 0) = k n 1 d d x y(x 0) = y' (x 0 ) = k 1 n Initial Conditions at x = x 0 y(x 0 ) = y(x 0 ) = k 0 IVP Higher Order ODEs (3A) 19

Initial Value Problem constant coefficients n +1 d n y a n d x + a d n 1 y n n 1 d x + + a d y n 1 1 d x + a 0 y = g(x) Linear Equation with constant coefficients n d n 1 d x n 1 y(x 0) = y (n 1) (x 0 ) = k n 1 n d d x y(x 0) = y' (x 0 ) = k 1 n Initial Conditions at x = x 0 y(x 0 ) = y(x 0 ) = k 0 y = c 1 e m 1 x + c 2 e m 2 x + + c n e m n x + y p ( x) n Parameters c i Higher Order ODEs (3A) 20

Boundary Value Problem a 2 (x) d2 y d x 2 + a 1(x) d y d x + a 0(x) y = g(x) y(a) = y 0 y' (a) = y 0 y(a) = y 0 y' (a) = y 0 Various Boundary Conditions y(b) = y 1 y(b) = y 1 y' (b) = y 1 y' (b) = y 1 Higher Order ODEs (3A) 21

1st Order 2nd Order IVP's d y d x = f ( x, y) on some interval I containing x 0 y(x 0 ) = k 1 Initial Condition 0 at x = x 0 (x 0, k 0 ) y(x) 1st Order IVP I d y d x = f ( x, y, y ') on some interval I containing x 0 (x 0, k 0 ) y(x) y(x 0 ) = k 2 Initial Conditions 0 at x = x 0 slope = k 1 y '( x 0 ) = k 1 2nd Order IVP I Higher Order ODEs (3A) 22

Existence of a unique solution : 1 st Order IVPs d y d x = f (x, y) on some interval I containing x 0 y(x) y(x 0 ) = k 1 Initial Condition 0 at x = x 0 (x 0, k 0 ) 1st Order IVP I f (x, y) and f y are continuous on R The solution y(x) of the IVP 1) exists on the interval I0 2) is unique R a x b I0 x 0 h x x 0 +h (h>0) c y d contained in [a, b] Higher Order ODEs (3A) 23

Existence of a unique solution : Linear 1 st Order IVPs a 1 ( x) d y d x + a Non-homogeneous 0 ( x) y = g( x) on some interval I containing x 0 Equation with variable coefficients y(x 0 ) = k 0 1 Initial Condition at x = x 0 1st Order IVP a 1 ( x), a 0 ( x), g( x) are all continuous on the interval I and a n (x) 0 The solution y(x) of the IVP 1) exists on the interval I 2) is unique Higher Order ODEs (3A) 24

Existence of a unique solution : Linear 1 st Order IVPs d y d x + p( x) y = g( x) Non-homogeneous on some interval I containing x 0 Equation with variable coefficients y(x 0 ) = k 0 1 Initial Condition at x = x 0 1st Order IVP p(x), g(x) are all continuous on the interval I The solution y(x) of the IVP 1) exists on the interval I 2) is unique Higher Order ODEs (3A) 25

Existence : Proof y ' + p( x) y = g(x) y(x 0 ) = k 0 p(x) continuous on the interval I x x 0 p(s)ds differentiable d dx x x 0 μ( x) = e x0 p(s) ds = p(x) x p( s)ds μ '( x) = e p( s)ds x 0 p( x) x (μ( x) y ) ' = μ '( x) y + μ (x) y' = μ(x) p(x) y + μ(x) y ' (μ( x) y ) ' = μ (x) g( x) x [μ(x) y ] x0 x = μ(s) g(s) ds x 0 μ( x) y(x) μ( x 0 ) y(x 0 ) = x x 0 μ( s) g(s)ds x μ( x) y(x) y(x 0 ) = μ(s) g( s)ds x 0 y(x) = 1 { x μ( x) y(x 0) + x 0 μ(s) g(s) ds} http://faculty.atu.edu/mfinan/3243/diffq1book.pdf Higher Order ODEs (3A) 26

Uniqueness : Proof y 1 ' + p( x) y 1 = g(x) y 1 ( x 0 ) = k 0 y 2 ' + p( x) y 2 = g(x) y 2 ( x 0 ) = k 0 w ( x) = y 1 (x) y 2 ( x) w ' + p( x) w = 0 μ( x) = e x 0 x p( s)ds μ(x) w ' + μ( x) p( x) w = 0 (μ (x)w)' = 0 w ( x) = C e x 0 x p(s) ds w ( x 0 ) = y 1 ( x 0 ) y 2 ( x 0 ) = k 0 k 0 = 0 C = 0 w ( x) = 0 μ( x) w( x) = C w ( x) = C /μ( x) http://faculty.atu.edu/mfinan/3243/diffq1book.pdf Higher Order ODEs (3A) 27

1 st Order IVP Counter examples (1) y ' = y y(0) = y 0 IVP f (x, y) = f ( y) = y continuous y > 0 y < 0 y ' = y y ' = y f y = d f d y discontinuous over any interval containing y = 0 1 y d y = d x ln y = x + c y = e x + c y = C e x 1 y d y = d x ln y = x + c y = e x + c y = C e x a unique solution for [y > 0], [y = 0], [y < 0] f (x, y) and f y are continuous on R The solution y(x) of the IVP 1) exists on the interval I0 2) is unique R a x b I0 x 0 h x x 0 +h (h>0) c y d contained in [a, b] Higher Order ODEs (3A) 28

1 st Order IVP Counter examples (2) y ' = y 1/3 y(0) = 0 IVP f (x, y) = f ( y) = y 1/3 continuous y 1/3 d y = d x y 2/3 = 2 3 x f y = d f d y = 1 3 y 2/3 discontinuous 3 2 y2/3 = x + c y = 0 c = 0 y 2 = ( 2 3 x ) 3 y = ± ( 2 3 x ) 3/2 two possible solutions + {y = 0} f (x, y) and f y are continuous on R The solution y(x) of the IVP 1) exists on the interval I0 2) is unique R a x b I0 x 0 h x x 0 +h (h>0) c y d contained in [a, b] Higher Order ODEs (3A) 29

1 st Order IVP Counter examples (3) y ' = y y(0) = y 0 IVP y ' = y 1/3 y(0) = 0 IVP f (x, y) = y f (x, y) = y 1 /3 f y discontinuous over any interval containing y = 0 f y discontinuous over any interval containing y = 0 y = C e x y = ± ( 2 y = C e x 3 x ) 3/2 a unique solution for [y > 0], [y = 0], [y < 0] non-unique solutions Higher Order ODEs (3A) 30

1 st Order IVP Counter examples (4) y ' = y y(0) = y 0 IVP y ' = y 1/3 y(0) = 0 IVP Higher Order ODEs (3A) 31

Direction Field of ( x/y) dy dx = x y 2-d version of F(x,y) F (x, y) = x y Higher Order ODEs (3A) 32

3-d Plot of ( x/y) dy dx = x y F (x, y) = x y 3-d plot of F(x,y) 1 y x Higher Order ODEs (3A) 33

Existence of a unique solution a n ( x) dn y d x n + a n 1(x) dn 1 y d x n 1 + + a 1(x) d y d x + a 0(x) y = g(x) Non-homogeneous Equation with variable coefficients d n 1 d x n 1 y(x 0) = k n 1 d d x y(x 0) = y' (x 0 ) = k 1 n Initial Conditions at x = x 0 y(x 0 ) = y(x 0 ) = k 0 IVP a n ( x), a n 1 (x), a 1 (x), a 0 ( x), are all continuous on the interval I and a n ( x) 0 g(x) The solution y(x) of the IVP 1) exists on the interval I 2) is unique Higher Order ODEs (3A) 34

Continuous Function a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be a "discontinuous function". A continuous function with a continuous inverse function is called a homeomorphism. Removable discontinuity Jump discontinuity Essential discontinuity Higher Order ODEs (3A) 35

Differentiable Function a differentiable function of one real variable is a function whose derivative exists at each point in its domain. the graph of a differentiable function must have a non-vertical tangent line at each point in its domain, be relatively smooth, and cannot contain any breaks, bends, or cusps. not differentiable at x=0 Higher Order ODEs (3A) 36 a differentiable function

Differentiability and Continuity If f is differentiable at a point x0, then f must also be continuous at x0. any differentiable function must be continuous at every point in its domain. The converse does not hold: a continuous function need not be differentiable. Differentiable Continuous not differentiable at x=0 Higher Order ODEs (3A) 37

Check for Linear Independent Solutions Homogeneous Linear n-th order differential equation a n (x) d n y d x + a n 1(x) dn 1 y n d x n 1 + + a 1 (x) d y d x + a 0(x) y = 0 n-th order Homogeneous y 1, y 2,, y n n linearly independent solutions W ( y 1, y 2,, y n ) 0 {y 1, y 2,, y n } fundamental set of solutions y = c 1 y 1 + c 2 y 2 + + c n y n general solution The general solution for a homogeneous linear n-th order differential equation Higher Order ODEs (3A) 38

References [1] http://en.wikipedia.org/ [2] M.L. Boas, Mathematical Methods in the Physical Sciences [3] E. Kreyszig, Advanced Engineering Mathematics [4] D. G. Zill, W. S. Wright, Advanced Engineering Mathematics [5] www.chem.arizona.edu/~salzmanr/480a