Definition. By a linear ODE of order 1 we mean any ODE written in the - PDF document

Definition. By a linear ODE of order 1 we mean any ODE written in the form y ′ + a ( x ) y = b ( x ), where a, b are some functions. This equation is called homogeneous if b ( x ) = 0. Given an ODE y ′ + a ( x ) y = b ( x ), by its associated homogeneous equation we mean the equation y ′ + a ( x ) y = 0. 1

(variation of parameter for linear ODE of order 1) Algorithm Given: equation y ′ + a ( x ) y = b ( x ). 1. Using separation, find a general solution y h of the associated homo- geneous equation y ′ + a ( x ) y = 0. It has the form y h ( x ) = C · u ( x ), which includes also stationary solutions. 2. Variation of parameter: Seek a solution of the form y ( x ) = C ( x ) · u ( x ). Either substitute this y ( x ) into the given equation y ′ + a ( x ) y = b ( x ) and cancel, or remember that it leads to the equation C ′ ( x ) u ( x ) = b ( x ). � b ( x ) Then C ( x ) = u ( x ) dx , substitute this C ( x ) into y ( x ) = C ( x ) u ( x ). 3. If you take for C ( x ) one particular antiderivative, then you get one particular solution y p ( x ), the general solution is then y = y p + y h . If you include “+ C ” when deriving C ( x ), then after substituting it into y ( x ) = C ( x ) u ( x ) you get the general solution. 2

(on solution of linear ODE of order 1) Theorem. Consider a linear ODE y ′ + a ( x ) y = b ( x ). Assume that a ( x ) , b ( x ) are continuous functions on an open interval I , let A be some antiderivative of a on I . Then the given equation has a solution on I of the form �� � b ( x ) e A ( x ) dx e − A ( x ) . If B is some antidrivative of b ( x ) e A ( x ) on I , then a general solution of the given equation on I is y ( x ) = ( B ( x ) + C ) e − A ( x ) . 3

(on structure of solution set of linear ODE of order 1) Theorem. Let y p be some particular solution of the equation y ′ + a ( x ) y = b ( x ) on an open interval I . A function y 0 ( x ) is a solution of this equation on I if and only if y 0 = y p + y h , where y h ( x ) is some solution of the associated homogeneous equation on I . 4

Definition. By a linear ordinary differential equation of order n (LODE) we mean any ODE of the form y ( n ) + a n − 1 ( x ) y ( n − 1) + . . . + a 1 ( x ) y ′ + a 0 ( x ) y = b ( x ), where a n − 1 , . . . , a 0 , b are some functions. This equation is called homogeneous if b ( x ) = 0. Given a linear ODE y ( n ) + a n − 1 ( x ) y ( n − 1) + . . . + a 1 ( x ) y ′ + a 0 ( x ) y = b ( x ), by its associated homogeneous equation we mean the equation y ( n ) + a n − 1 ( x ) y ( n − 1) + . . . + a 1 ( x ) y ′ + a 0 ( x ) y = 0. 5

(on existence and uniqueness for LODE) Theorem. Consider a linear ODE y ( n ) + a n − 1 ( x ) y ( n − 1) + . . . + a 1 ( x ) y ′ + a 0 ( x ) y = b ( x ). (L) If a n − 1 , . . . , a 0 , b are continuous on an open interval I , then for all x 0 ∈ I and y 0 , y 1 , . . . , y n − 1 ∈ I R there exists a solution to the IVP (L), y ( x 0 ) = y 0 , y ′ ( x 0 ) = y 1 , . . . , y ( n − 1) ( x 0 ) = y n − 1 on I and it is unique there. 6

(on structure of solution set of LODE) Theorem. Consider a homogeneous linear ODE y ( n ) + a n − 1 ( x ) y ( n − 1) + . . . + a 1 ( x ) y ′ + a 0 ( x ) y = 0. If a i are continuous on an open interval I , then the set of all solutions of this equation on I is a linear space of dimension n . 7

Definition. Consider a homogeneous linear ODE y ( n ) + a n − 1 ( x ) y ( n − 1) + . . . + a 1 ( x ) y ′ + a 0 ( x ) y = 0. Assume that a i are continuous on an open interval I . By a fundamental system of solutions of this equation on I we mean any basis of the space of all solutions of this equation on I . 8

Definition. Let y 1 , y 2 , . . . , y n be ( n − 1)-times differentiable functions. We define their Wronskian as � � y 1 ( x ) y 2 ( x ) y n ( x ) . . . � � � � y ′ 1 ( x ) y ′ 2 ( x ) y ′ n ( x ) . . . � � � � W ( x ) = . . . . � � . . . . . . � � � � y ( n − 1) y ( n − 1) y ( n − 1) ( x ) ( x ) . . . ( x ) n 1 2 9

Theorem. Consider a homogeneous linear ODE y ( n ) + a n − 1 ( x ) y ( n − 1) + . . . + a 1 ( x ) y ′ + a 0 ( x ) y = 0), Let a i be continuous on an open interval I . Let y 1 , y 2 , . . . , y n be solutions of this equation on I , let W be their Wronskian. These functions form a linearly independent set (and thus a fundamental system) if and only if W ( x ) � = 0 on I , which is if and only if W ( x 0 ) � = 0 for some x 0 ∈ I . 10

Definition. By a linear ODE with constant coefficients we mean any linear ODE for which a 0 ( x ) = a 0 , a 1 ( x ) = a 1 , . . . , a n − 1 ( x ) = a n − 1 are con- stant functions. 11

Definition. Consider a homogeneous linear ODE with constant coefficients y ( n ) + a n − 1 y ( n − 1) + . . . + a 1 y ′ + a 0 y = 0. We define its characteristic polynomial as p ( λ ) = λ n + a n − 1 λ n − 1 + . . . + a 1 λ + a 0 . We define its characteristic equation as p ( λ ) = 0. The solutions of this equation are called characteristic numbers or eigenvalues of the given ODE. 12

Fact. Consider a homogeneous linear ODE with constant coefficients y ( n ) + a n − 1 y ( n − 1) + . . . + a 1 y ′ + a 0 y = 0. Let λ 0 be its characteristic number. Then y ( x ) = e λ 0 x is a solution of this equation. If λ 1 , . . . , λ N are distinct characteristic numbers of this equation, then { e λ 1 x , . . . , e λ N x } is a linearly independent set of solutions. 13

Fact. Consider a homogeneous linear ODE with constant coefficients y ( n ) + a n − 1 y ( n − 1) + . . . + a 1 y ′ + a 0 y = 0. Let λ 0 be its characteristic number with multiplicity m . Then e λ 0 x , x e λ 0 x , . . . , x m − 1 e λ 0 x are solutions of this equation and they form a linarly independent set. 14

(on fundamental system for LODE) Theorem. Consider a homogeneous linear ODE with constant coefficients y ( n ) + a n − 1 y ( n − 1) + . . . + a 1 y ′ + a 0 y = 0. Let λ be its characteristic number of multiplicity m . R , then e αx , x e αx , . . . , x m − 1 e αx are solutions of the (1) If λ = α ∈ I associated homogeneous equation on I R and they are linearly indepen- dent. C , β � = 0, then e αx sin( βx ), x e αx sin( βx ), . . . , (2) If λ = α ± βj ∈ I x m − 1 e αx sin( βx ), e αx cos( βx ), x e αx cos( βx ), . . . , x m − 1 e αx cos( βx ) are solutions of the associated homogeneous equation on I R and they are linearly independent. (3) The set of functions from (1) and (2) for all characteristic numbers is linearly independent and it forms a fundamental system of the given equation on I R . 15

(on structure of solution set of linear ODE) Theorem. Let y p be some particular solution of a given linear ODE on an open interval I . A function y 0 is a solution of this equation on I if and only if y 0 = y p + y h for some solution y h of the associated homogeneous equation on I . Consequently, if y h is a general solution of the associated homogeneous equation on I , then y p + y h is a general solution of the given equation. 16

( guessing a solution for special right hand-side) Theorem. Consider a linear ODE with constant coefficients y ( n ) + a n − 1 y ( n − 1) + . . . + a 1 y ′ + a 0 y = b ( x ) . Assume that b ( x ) = e αx [ P ( x ) sin( βx ) + Q ( x ) cos( βx )] for some polyno- mials P, Q , denote d = max(deg( P ) , deg( q )). Let k be the multiplicity of the number α ± βj as a characteristic number of the given equation (we put k = 0 if it is not a char. no. at all). Then there are polynomials � P, � Q of degree at most d such that y ( x ) = x k e αx [ � P ( x ) sin( βx ) + � Q ( x ) cos( βx )] is a solution of the given equation on I R . This is called the method of undetermined coefficients. 17

Simpler forms of incomplete right hand-sides: ⇒ y ( x ) = x k � • b ( x ) = P ( x ) = P ( x ), where k is the multiplicity of 0; • b ( x ) = P ( x ) e αx = ⇒ y ( x ) = x k � P ( x ) e αx , where k is the multiplicity of α ; • b ( x ) = P ( x ) sin( βx ) + Q ( x ) cos( βx ) = ⇒ y ( x ) = x k [ � P ( x ) sin( βx ) + � Q ( x ) cos( βx )], where k is the multiplicity of 0 ± βj . 18

( superposition principle ) Theorem. Consider a linear ODE with left hand-side L ( y ) = y ( n ) + a n − 1 ( x ) y ( n − 1) + . . . + a 1 ( x ) y ′ + a 0 ( x ) y . Let y 1 be a solution of L ( y ) = b 1 ( x ) on an open interval I and y 2 be a solution of L ( y ) = b 2 ( x ) on I . Then y 1 + y 2 is a solution of L ( y ) = b 1 ( x ) + b 2 ( x ) on I . 19