Sec_2.5 - 2.5 Linear Equations Definition 2.5 Linear...

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2.5 Linear Equations Definition 2.5 Linear Equation A differential equation of the form ( 29 ( 29 ( 29 1 0 dy a x a x y g x dx + = is said to be a linear equation. Integrating Factor : ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 { } ( 29 ( 29 ( 29 ( 29 0 1 0 1 1 1 0 0 0 . ln pdx a x g x dy dy a x a x y g x y dx dx a x a x dy dy p x y f x p x y f x dx dx dy p x y f x dx x dy x p x y f x dx It is an exact differential equation x p x y f x x y x x p x x x d pdx pdx c ce Ch μ μ μ μ μ μ μ μ μ μ + = + = + = + - = + - = + - = ° ° - = ° ° = ° = = + ° = 1 . pdx oose c e μ = = Ex. Solve 6 4 x dy x y x e dx - = . Solution : ( 29 ( 29 ( 29 6 5 4 4ln 4 4 4 4 . , int . x x dx x x dy dy x y x e y x e dx dx x p x So the egrating factor x x e x e x μ μ μ - - - - = - = = - ° = = =
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( 29 ( 29 4 5 4 4 4 5 4 4 4 . x x x x x x x dy d x x y xe x y xe dx dx d x y xe dx x y xe e c y x e x e cx - - - - - - = = = = - + = - + Ex. Solve 3 0 dy y dx - = .
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