## FANDOM

17 Pages

A technique to solve first order differential equations of the form

${{dy \over dx} + p(x)y = q(x)}$.

• Calculate the integration factor ${\rho (x) = e^{\int p(x) dx}}$.
• Multiply each side of the equation by ${\rho (x)}$
• Integrate both sides of the equation to get ${\rho(x)y(x) = \int \rho(x)q(x) dx}$.
• Sove for y(x).

## Example

${{dy \over dx} - y = {11 \over 8}e^{-x/3}}$

${\Rightarrow \rho(x) = e^{\int (-1) dx} = e^{-x}}$

${\Rightarrow e^{-x}{dy \over dx} - e^{-x}y = {11 \over 8}e^{-4x/3}}$

${\Rightarrow e^{-x}y = \int {11 \over 8}e^{-4x/3} dx = {-33 \over 32} e^{(-4x/3)} + C}$

${\Rightarrow y(x) = {-33 \over 32} e^{(-x/3)} + Ce^{x}}$