First-Order Linear Equations

Author: John J Weber III, PhD Corresponding Textbook Sections:

Expected Educational Results

Linear Equations

Method for Solving First-Order ODEs

  1. Identify if the DE is a linear first-order DE;

  2. Rewrite the DE in standard form, dydx+P(x)y=Q(x);

  3. Compute the integrating factor: μ(x)=eP(x)dx [no +C needed here];

  4. Multiply both sides of the DE in standard form by: μ(x) to obtain μ(x)dydx+μ(x)P(x)y=μ(x)Q(x)

  5. The above result is equivalent to, by Chain Rule for derivatives: ddx[μ(x)y]=μ(x)Q(x)

  6. Integrate both sides of the DE: ddx[μ(x)y]=μ(x)Q(x)dx

  7. The above is equivalent to, by FTC-1: μ(x)y=μ(x)Q(x)dx

  8. Divide both sides of above equation: y=1μ(x)μ(x)Q(x)dx

  9. Solve for y, i.e., find an explicit solution, if possible. If it is not possible to solve for y, leave the answer as an implicit solution;

  10. If there is an initial condition, y(x0)=y0, use it to find the constant, C;

  11. Check for any missing solutions.

NOTE: If the solution to an ODE contains an integral that does not have an analytical antiderivative, then write the solution as an integral equation.

CC BY-NC-SA 4.0

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

Last Modified: Wednesday, 2 September 2020 07:22 EDT