Methods from Second-order constant-coefficient equations

Solve the characteristic equation and choose the real homogeneous modes according to the roots:

• If $r_1\neq r_2$ are distinct real roots, use $$y_h=Ae^{r_1x}+Be^{r_2x}.$$
• If $r$ is a repeated real root, use $$y_h=(A+Bx)e^{rx}.$$
• If the roots are $r=\alpha\pm i\beta$ with $\beta\neq0$, use $$y_h=e^{\alpha x} \bigl(A\cos(\beta x)+B\sin(\beta x)\bigr).$$

For suitable forcing functions:

• a polynomial of degree $n$ suggests a general polynomial of degree $n$;
• $e^{\beta x}$ suggests $Ce^{\beta x}$;
• $p_n(x)e^{\beta x}$ suggests $q_n(x)e^{\beta x}$, where $q_n$ has degree $n$;
• $\sin(\beta x)$ or $\cos(\beta x)$ suggests $$C_1\cos(\beta x)+C_2\sin(\beta x);$$
• $e^{\alpha x}\sin(\beta x)$ or $e^{\alpha x}\cos(\beta x)$ suggests $$e^{\alpha x} \bigl(C_1\cos(\beta x)+C_2\sin(\beta x)\bigr).$$

Substitute the trial into the equation and match coefficients. If the trial overlaps with a homogeneous mode whose characteristic root has multiplicity $s$, multiply the whole trial by $x^s$ (so a simple overlap requires one factor of $x$, while a repeated root requires $x^2$).

Differentiate an integral equation with a variable limit, record the initial values obtained at the endpoint and solve the resulting differential equation subject to those values. Finally, check the original integral equation because differentiation can enlarge the solution set.