Solving Differential Equations by Integrating Factors
Solving Differential Equations by Integrating Factors:

Integrating Factor Method Overview:
 Applicable to firstorder linear ordinary differential equations that are not exact.
 The goal is to find a suitable integrating factor to transform the equation into an exact form.

Steps to Solve by Integrating Factors:
a. Identify the Equation: Start with a firstorder linear differential equation of the form $\frac{dy}{dx}+P(x)y=Q(x)$.
b. Determine the Integrating Factor: The integrating factor $\mu (x)$ is obtained by $\mu ={e}^{\int P(x)\text{\hspace{0.17em}}dx}$.
c. Multiply and Solve: Multiply both sides of the equation by the integrating factor $\mu (x)$.
 This step makes the lefthand side of the equation exact.
 The equation transforms into $\mu (x)\frac{dy}{dx}+\mu (x)P(x)y=\mu (x)Q(x)$.
d. Integrate the Exact Equation: The equation becomes exact after multiplication by the integrating factor. Integrate both sides with respect to $x$ to find the solution.

Example of Solving a Differential Equation by Integrating Factor:
Consider the differential equation $\frac{dy}{dx}+2xy=x$.
a. Identify the Equation: $\frac{dy}{dx}+2xy=x$.
b. Determine the Integrating Factor: $\mu ={e}^{\int 2x\text{\hspace{0.17em}}dx}={e}^{{x}^{2}}$.
c. Multiply the Equation by the Integrating Factor:
 Multiply both sides by ${e}^{{x}^{2}}$: ${e}^{{x}^{2}}\frac{dy}{dx}+2xy{e}^{{x}^{2}}=x{e}^{{x}^{2}}$.
d. Integrate the Transformed Equation:

This step makes the left side exact.

Integrate both sides with respect to $x$: $\int {e}^{{x}^{2}}\frac{dy}{dx}\text{\hspace{0.17em}}dx+\int 2xy{e}^{{x}^{2}}\text{\hspace{0.17em}}dx=\int x{e}^{{x}^{2}}\text{\hspace{0.17em}}dx$

Simplify and solve the equation to obtain the solution for $y$.