Undetermined coefficients

The general solution of \(ay''+by'+cy=g(t)\) is \(y=y_h+y_p\): the homogeneous solution plus any particular solution. Differentiating a polynomial, an exponential, or a sine or cosine produces functions of the same type, so when \(g\) is of that class we guess a \(y_p\) of the same form with unknown coefficients \(A_i\); substituting and matching like terms leaves a linear system for them. For \(g=8t\) guess \(At+B\); for \(g=10\sin t\), \(A\cos t+B\sin t\), because differentiating one brings in the other.

The exception is resonance. If the guess contains a term that already solves the homogeneous equation, substituting gives zero and there is no way to match \(g\). The remedy is to multiply by \(t^{s}\), where \(s\) is the multiplicity of \(k\) (or of \(k\pm i\beta\)) as a root of the characteristic equation. For \(y''-2y'-3y=e^{3t}\), with roots \(3\) and \(-1\), guess \(Ate^{3t}\) and get \(A=\tfrac14\); for a double root you need \(t^2\).

When \(g\) is a sum, solve each term separately and add the particular solutions (superposition). The method fails for \(\tan t\), \(\ln t\) or \(1/t\), whose derivatives keep producing new functions; there variation of parameters is used. Initial conditions are imposed at the end, on \(y=y_h+y_p\), never on \(y_p\) alone.