Solving Differential Equations by Separation of Variables

Solving Differential Equations by Separation of Variables:

  1. Method Overview:

    • Applicable to certain first-order ordinary differential equations (ODEs) that can be written in the form dydx=f(x)g(y) or a similar separable form.
    • Involves isolating variables (putting all y terms on one side and all x terms on the other) before integrating both sides separately.
  2. Steps to Solve by Separation of Variables:

    a. Identify a Separable Form: Ensure the differential equation can be written in the form dydx=f(x)g(y) or similar.

    b. Isolate Variables: Rearrange the equation to have all y terms on one side and all x terms on the other side.

    c. Integrate Both Sides: Integrate both sides of the equation with respect to their respective variables.

    d. Add Constant of Integration: Include a constant of integration when integrating each side of the equation.

  3. Example: Solving a Differential Equation by Separation of Variables:

    Consider the equation dydx=x2y.

    a. Separate Variables: Rewrite the equation by isolating variables: dyy=x2dx

    b. Integrate Both Sides:

    • Integrate the left side with respect to y and the right side with respect to x: 1ydy=x2dx
    • This yields lny=x33+C1, where C1 is the constant of integration.

    c. Solve for y:

    • Exponentiate both sides to solve for y: y=ex33+C1=eC1ex33
    • y=Cex33 where C=±eC1.

    d. Final Result:

    • The solution to the differential equation dydx=x2y by separation of variables is y=Cex33, where C is an arbitrary constant.