Solving Differential Equations by Separation of Variables
Solving Differential Equations by Separation of Variables:
- Applicable to certain first-order ordinary differential equations (ODEs) that can be written in the form or a similar separable form.
- Involves isolating variables (putting all terms on one side and all terms on the other) before integrating both sides separately.
Steps to Solve by Separation of Variables:
a. Identify a Separable Form: Ensure the differential equation can be written in the form or similar.
b. Isolate Variables: Rearrange the equation to have all terms on one side and all 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.
Example: Solving a Differential Equation by Separation of Variables:
Consider the equation .
a. Separate Variables: Rewrite the equation by isolating variables:
b. Integrate Both Sides:
- Integrate the left side with respect to and the right side with respect to :
- This yields , where is the constant of integration.
c. Solve for :
- Exponentiate both sides to solve for :
- where .
d. Final Result:
- The solution to the differential equation by separation of variables is , where is an arbitrary constant.