1. Homogeneous: ay'' + by' + cy = 0
Characteristic equation: ar² + br + c = 0
Discriminant: D = b² - 4ac
| Case | Roots | General Solution |
|---|
| D > 0 | r1, r2 real distinct | c1er1t + c2er2t |
| D = 0 | r repeated | (c1 + c2t)ert |
| D < 0 | α ± βi | eαt(c1cos βt + c2sin βt) |
Wronskian: W = y1y2' - y2y1'. If W ≠ 0, solutions are linearly independent.
2. Nonhomogeneous: ay'' + by' + cy = g(t)
General solution: y = yh + yp (homogeneous + particular).
Undetermined Coefficients
| g(t) | Guess for yp |
|---|
| Polynomial degree n | Antn + ... + A0 |
| eαt | Aeαt |
| cos βt or sin βt | A cos βt + B sin βt |
| eαtcos βt | eαt(A cos βt + B sin βt) |
Modification Rule: If guess overlaps yh, multiply by t (or t² for double overlap).
Variation of Parameters
yp = -y1∫(y2g/W)dt + y2∫(y1g/W)dt
Must be in standard form (leading coeff = 1). Works for ANY g(t).