For constant coefficients and forcing built from polynomials, exponentials, sines and cosines, a particular solution has the same shape as the forcing, with unknown coefficients. Substitute the guess, match coefficients, and solve a small linear system. The one trap is a guess that already solves the homogeneous equation: it can then only produce zero, never the forcing, and multiplying the guess by x repairs it.

Pushing a swing in a steady rhythm. After a while it moves in your rhythm, not its own. The response to a sinusoidal push is a sinusoid of the same frequency, and only its size and its timing are unknown — those are the undetermined coefficients.

For $y''+y'+y=\cos2x$ the particular solution is $-\tfrac{3}{13}\cos2x+\tfrac{2}{13}\sin2x$: the same frequency as the forcing, a smaller amplitude and a shifted phase. That is why the guess needs both a cosine and a sine.

For $ay''+by'+cy=g(x)$ with a polynomial, an exponential , a sine or cosine of , or a product of these, try a particular solution of the same shape with unknown coefficients. If any term of the guess solves the homogeneous equation, multiply the whole guess by — by if that still does.

• $y''-y=3e^{2x}$: try $Ae^{2x}$; gives , so .

The computation of the repeated-roots lesson holds for any k: applied to x times the exponential, the operator gives the characteristic polynomial at k times x times the exponential, plus its derivative at k times the exponential. When k is a simple root the first term vanishes and the second does not, so the modified guess produces a multiple of the exponential, and dividing by that multiple matches the forcing.

The computation of the repeated-roots lesson, for any k.

At a simple root the polynomial vanishes and its derivative does not.

So the modified guess produces a multiple of the exponential.

Matching the forcing fixes the coefficient.

Guess a particular solution shaped like the forcing, with unknown coefficients, and let the equation fix them.

Great work! You now know how to:

• choose the shape of a particular solution from the shape of the forcing
• find its coefficients by matching
• recognise when the guess solves the homogeneous equation and multiply by x
• say which forcing terms the method cannot handle