Nonhomogeneous Equations and Particular Solutions
The full solution is the homogeneous solution plus any particular one. Undetermined coefficients guesses the shape of the forcing; variation of parameters handles the rest.
What a learner can do afterwards
- Choose a trial solution matching the forcing term
- Adjust the trial when it duplicates a homogeneous solution
- Use variation of parameters when no simple guess applies
1 · Read
A forcing term on the right side changes the problem: the full solution is the homogeneous answer plus one particular guest that matches the forcing. Undetermined coefficients guesses that guest from a menu. Exponentials invite exponentials, polynomials invite polynomials, and sines invite sine plus cosine, each with unknown capital letter coefficients to solve for. You cannot pick the guest without first solving the homogeneous equation.
Solve y double prime minus y equals e to the x. The characteristic roots are 1 and minus 1. The obvious guess A e to the x duplicates a homogeneous solution, so multiply by x until it is fresh: the trial becomes A x e to the x. Compare with y double prime plus 3 y prime plus 2 y equals 5 e to the 3 x, where the roots are minus 1 and minus 2: rate 3 duplicates nothing, so the plain trial A e to the 3 x works untouched.
Duplication is the one trap, and the homogeneous roots tell you exactly which guesses clash. Add an x only when there is a clash: an x the problem does not need drops the plain term, and the trial can no longer fit. When no menu item fits at all, variation of parameters always works by letting the constants vary, at the price of messier integrals. It asks only for the homogeneous solutions, so keep them handy.
Run the checklist on every problem: split into homogeneous plus particular, choose the trial by the shape of the forcing, adjust with x on duplication, and fall back to variation of parameters. And match the tool to the problem: undetermined coefficients needs constant coefficients with menu shaped forcing, while variation of parameters is general.
Solve the homogeneous part first, guess the forcing shape, multiply by x on clashes, and vary parameters as backup.
Readings 'The guest that matches the forcing', 'Roots first' and 'A checklist for every problem' are original to this dataset. All 5 extra questions are original to this dataset.
2 · Watch
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13 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.