Past exam of the mathematical methods I course of the University of Oxford / 2022 / 11 / c

The exponential substitution removing a constant first derivative, $y=e^{x}u$, reduces the equation to $u^{\prime \prime }+{\beta }^{2}u=1+\cos x$. For $\beta \ne 0$ and ${\beta }^2\ne 1$, solve the homogeneous harmonic equation and use a constant plus a cosine for the particular solution:There is a further exceptional case inside the printed condition ${\beta }^2\ne 1$: $\beta =0$ must be treated separately. It gives $u^{\prime \prime }=1+\cos x$, henceThis retains the constant forcing's resonance with the zero-frequency homogeneous equation. Substitution verifies both branches.

For ${\beta }^2=1$, the reduced equation is $u^{\prime \prime }+u=1+\cos x$. The constant forcing has particular solution one, while the resonant cosine requires $u_p=(x/2)\sin x$. ThusThe secular factor $x$ is the signature of resonance. It must replace the divergent nonresonant coefficient $1/({\beta }^2-1)$; substituting ${\beta }^2=1$ directly into that coefficient is invalid.