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 dydx+P(x)y=Q(x).

    b. Determine the Integrating Factor: The integrating factor μ(x) is obtained by μ=eP(x)dx.

    c. Multiply and Solve: Multiply both sides of the equation by the integrating factor μ(x).

    • This step makes the left-hand side of the equation exact.
    • The equation transforms into μ(x)dydx+μ(x)P(x)y=μ(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.

  3. Example of Solving a Differential Equation by Integrating Factor:

    Consider the differential equation dydx+2xy=x.

    a. Identify the Equation: dydx+2xy=x.

    b. Determine the Integrating Factor: μ=e2xdx=ex2.

    c. Multiply the Equation by the Integrating Factor:

    • Multiply both sides by ex2: ex2dydx+2xyex2=xex2.

    d. Integrate the Transformed Equation:

    • This step makes the left side exact.

    • Integrate both sides with respect to x: ex2dydxdx+2xyex2dx=xex2dx

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