Find a forced solution and check for resonance
For a nonhomogeneous equation, write y = y_c + y_p. The complementary part solves the homogeneous equation. Choose a particular trial matching the forcing, then substitute it to determine the unknown coefficients. If the trial already appears in the complementary solution, multiply it by enough powers of x to make it independent. This overlap is the algebraic signal of resonance in an undamped oscillator.
Worked practice example
Solve y″ + 4y = 8 with y(0) = 0 and y′(0) = 0.
| Step | Working |
|---|---|
| Complementary part | The roots are ±2i, so y_c = C₁cos(2x) + C₂sin(2x). |
| Constant particular trial | Let y_p = A. Substitution gives 4A = 8, so A = 2. |
| Apply y(0) = 0 | C₁ + 2 = 0, so C₁ = −2. |
| Apply y′(0) = 0 | 2C₂ = 0, so y = 2 − 2cos(2x). |
Check your method
Substitution gives y″ = 8cos(2x) and 4y = 8 − 8cos(2x), whose sum is 8. If the forcing had been cos(2x), a plain sine/cosine trial would overlap with y_c and would need an x multiplier. The constant forcing used here is not resonant.
Try it yourself
Find one particular solution of y″ + 4y = cos(2x).
Show the answer and method
Use y_p = Ax sin(2x). Then y_p″ + 4y_p = 4A cos(2x), giving A = 1/4. Thus y_p = x sin(2x)/4.
This written practice example was prepared for Jemshah revision. It is additional practice, not an official examination question or a transcript of the video.