Differential Equations · step 8 of 14

Undetermined coefficients

For a linear ODE with constant coefficients and a "nice" forcing term, the particular solution has the same shape as the forcing: a polynomial for a polynomial, Ae^(kx) for an exponential, and a combination of sine and cosine for either. The one complication is resonance — if your guess already solves the homogeneous equation, multiply it by x.

Practise undetermined coefficients free → All Differential Equations subtopics

✓ Unlimited questions · marked criterion by criterion · no card needed

Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Solve the homogeneous equation firstYou need the complementary function both for the final answer and to detect resonance.
Choose the trial form from the forcing termPolynomial of degree n → general degree-n polynomial. e^(kx) → Ae^(kx). sin or cos of ωx → A cos ωx + B sin ωx (BOTH terms, always).
Multiply by x if it duplicates a homogeneous solutionRepeat if necessary — multiply by x² for a double root.
Substitute and match coefficientsEquate coefficients of each independent function to get simultaneous equations for the unknowns.

Worked example

Find a particular solution of y″ − y = 3e^(2x).

Homogeneous solutions are e^x and e^(−x); e^(2x) is not among them, so no resonance.
Trial: y_p = Ae^(2x). Then y_p″ = 4Ae^(2x).
Substitute: 4Ae^(2x) − Ae^(2x) = 3Ae^(2x), and this must equal 3e^(2x).
So 3A = 3, giving A = 1.

Answer. y_p = e^(2x), and the general solution is y = Ae^x + Be^(−x) + e^(2x).

Where marks get dropped

These are the specific errors that cost credit on undetermined coefficients questions — QED's rubric penalises each of them separately.

Omitting the sine when the forcing is a cosine. Differentiation mixes them, so the trial needs both A cos ωx and B sin ωx.
Missing resonance. If e^(kx) already solves the homogeneous equation, the trial Ae^(kx) yields 0 = forcing, an impossible equation.
Using an incomplete polynomial trial. For forcing x², try Ax² + Bx + C, not just Ax².

Practise this until it is automatic

Unlimited fresh questions

QED generates new undetermined coefficients problems on demand at warm-up, exam and challenge level, so you can drill this one skill until it stops costing you marks.

Marked like an examiner

Every answer is scored against a point-by-point rubric with partial credit, so you see exactly which step of the method broke down — not just a tick or a cross.

Answer in real notation

A one-tap symbol palette, a visual equation editor and a truth-table builder — or photograph your handwritten working and QED converts it to LaTeX.

Saved to your library

Every question you generate is kept and re-takeable as a timed exam, and your Differential Equations mastery is tracked so you know when this is exam-ready.

Undetermined coefficients — frequently asked questions

What if the forcing term is a product?

Multiply the trial forms: for xe^(2x) try (Ax + B)e^(2x). For e^x sin 2x try e^x(A cos 2x + B sin 2x).

When does the method fail?

For forcing terms like tan x or 1/x whose derivatives never close into a finite family. Then use variation of parameters.

How do I detect resonance quickly?

Compare the exponent or frequency of the forcing with the characteristic roots. A match means multiply the trial by x.

The rest of Differential Equations

Solving and modelling with ODEs, from separable to systems. Each subtopic below has its own method, worked example and mark-losing traps.

Ready to make undetermined coefficients exam-proof?

Generate your first questions free — no card, no setup, no personal data stored. Practise until the method is second nature.

Start practising free →