# Solving Differential Equations by Integrating Factors

### Solving Differential Equations by Integrating Factors:

1. Integrating Factor Method Overview:

• Applicable to first-order 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.
2. Steps to Solve by Integrating Factors:

a. Identify the Equation: Start with a first-order linear differential equation of the form $\frac{dy}{dx}+P\left(x\right)y=Q\left(x\right)$.

b. Determine the Integrating Factor: The integrating factor $\mu \left(x\right)$ is obtained by $\mu ={e}^{\int P\left(x\right)\text{\hspace{0.17em}}dx}$.

c. Multiply and Solve: Multiply both sides of the equation by the integrating factor $\mu \left(x\right)$.

• This step makes the left-hand side of the equation exact.
• The equation transforms into $\mu \left(x\right)\frac{dy}{dx}+\mu \left(x\right)P\left(x\right)y=\mu \left(x\right)Q\left(x\right)$.

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.

3. 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 $y$.