Solving Differential Equations by Integrating Factors
Solving Differential Equations by Integrating Factors:
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.
Steps to Solve by Integrating Factors:
a. Identify the Equation: Start with a first-order linear differential equation of the form .
b. Determine the Integrating Factor: The integrating factor is obtained by .
c. Multiply and Solve: Multiply both sides of the equation by the integrating factor .
- This step makes the left-hand side of the equation exact.
- The equation transforms into .
d. Integrate the Exact Equation: The equation becomes exact after multiplication by the integrating factor. Integrate both sides with respect to to find the solution.
Example of Solving a Differential Equation by Integrating Factor:
Consider the differential equation .
a. Identify the Equation: .
b. Determine the Integrating Factor: .
c. Multiply the Equation by the Integrating Factor:
- Multiply both sides by : .
d. Integrate the Transformed Equation:
This step makes the left side exact.
Integrate both sides with respect to :
Simplify and solve the equation to obtain the solution for .