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How to Use The Integrating Factor Method for Differential Equations

The order of a differential equation is determined by the highest order derivative involved! Therefore, first order differential equations consist only of a function and its derivative.

A first order differential equation can be written as

y + f(x)y = g(x)

You solve these types of equations using integrating factors.

Theory

Integrating Factor

The integrating factor is eF(x), where

F(x) =f(x)dx

and F(x) = f(x) without C.

By using the product rule, you see that

(eF(x)y) = eF(x)y + f(x)eF(x)y

This is what you want to introduce into the original equation. Then the differential equation can be solved by integrating factors.

Rule

Instructions for Using the Integrating Factor to Solve Differential Equations

y + f(x)y = g(x)| eF(x) eF(x)y + f(x)eF(x)y = (eF(x)y) = eF(x)g(x) (eF(x)y)dx =eF(x)g(x)dx eF(x)y =eF(x)g(x)dx y =eF(x)g(x)dx eF(x)

y + f(x)y = g(x)| eF(x) eF(x)y + f(x)eF(x)y = (eF(x)y) = eF(x)g(x) (eF(x)y)dx =eF(x)g(x)dx eF(x)y =eF(x)g(x)dx y =eF(x)g(x)dx eF(x)

Example 1

Solve the differential equation y + 2xy = ex by using integrating factors

y + 2xy = ex | ex2 ex2y + 2xex2y = 2xex2 (ex2y) = 2xex2 (ex2y)dx = 2xex2dx ex2y = 2xex2dx ex2y = 2xeu 1 2xdu ex2y =eudu ex2y = eu + C ex2y = ex2 + C | ex2 y = 1 + Cex2

y + 2xy = ex | ex2 ex2y + 2xex2y = 2xex2 (ex2y) = 2xex2 (ex2y)dx = 2xex2dx ex2y = 2xex2dx ex2y = 2xeu 1 2xdu ex2y =eudu ex2y = eu + C ex2y = ex2 + C | ex2 y = 1 + Cex2

*

u = x2 du dx = 2x du = 2xdx 1 2xdu = dx

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