PP 10.5 - Linear Equations Definition A first-order linear...

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Unformatted text preview: Linear Equations Definition A first-order linear equation is an equation that can be put into the form ) ( ) ( x Q y x P dx dy = + P and Q are continuous functions over some interval . , 1 : Examples 2 = + ′ = + x xy y x e xy dx dy x The tool behind the solution method for linear equations is the product rule. [ ] + = ) ( ) ( ) ( ) ( ) ( ) ( x g dx d x f x g x f dx d x g x f dx d By multiplying both sides of a linear equation by an appropriate function (an integrating factor ), the derivative of a product is obtained on the left side of the equation. Integrating both sides of the equation (using the Fundamental Theorem of Calculus) yields a solution. ∫ = dx x P e x I ) ( ) ( is factor g integratin An ) ( ) ( x Q y x P dx dy = + Example: Solve the differential equation 2 1 = + y x dx dy ∫ = ⇒ = dx x e x I x x P 1 ) ( 1 ) ( x e x I x = = ⇒ ) ln( ) ( x xy dx d x y dx dy x x y x dx dy x 2 ) ( 2 2 1 = ⇒ = + ⇒ =...
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PP 10.5 - Linear Equations Definition A first-order linear...

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