This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Linear Equations Definition A firstorder 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 = ⇒ = + ⇒ =...
View
Full
Document
This note was uploaded on 01/21/2011 for the course PHYS 4A 60865 taught by Professor L. oldewurtel during the Fall '09 term at Irvine Valley College.
 Fall '09
 L. OLDEWURTEL

Click to edit the document details