Module 4: Second-Order Linear ODEs
Learn to solve the most important class of differential equations in science and engineering: second-order linear ODEs with constant coefficients, both homogeneous and nonhomogeneous.
Your Progress
Learning Objectives
By the end of this module, you will be able to:
- Solve homogeneous second-order linear ODEs with constant coefficients using the characteristic equation
- Handle all three cases: real distinct, repeated, and complex conjugate roots
- Find particular solutions using the method of undetermined coefficients
- Apply variation of parameters to find particular solutions for any forcing function
- Combine homogeneous and particular solutions for the complete general solution
Module Lessons
Homogeneous with Constant Coefficients
Set up ay'' + by' + cy = 0 and find solutions using the characteristic equation ar² + br + c = 0.
35-45 minutes
Characteristic Equation: Real, Repeated, Complex Roots
Master all three cases of the characteristic equation and write the general solution for each.
40-50 minutes
Undetermined Coefficients
Find particular solutions by guessing forms for polynomial, exponential, and trigonometric forcing.
40-50 minutes
Variation of Parameters
Use the Wronskian-based formula to handle any forcing function, including ones undetermined coefficients cannot.
35-45 minutes
After the Lessons
Practice Problems
10 problems covering all four methods for second-order linear ODEs.
Practice Problems