6.38 Summary of method

To solve the equation $ay^{\prime\prime}+by^{\prime}+cy=q(x)$:

(a) First solve the corresponding homogeneous equation to obtain the complementary factor (CF);

(b) If $q(x)$ is not an exceptional case then the PI has the same form as $q(x)$, i.e.:

- if $q(x)$ is a polynomial of degree $n$ then so is the PI;

- if $q(x)$ is of the form $Ae^{\kappa x}$ then so is the PI;

- if $q(x)$ is of the form $(C\cos\kappa x+D\sin\kappa x)e^{\lambda x}$ then so is the PI (both terms needed).

(c) For an exceptional case, the PI has the form of $q(x)$ multiplied by $x$, or in the case of a double root, the form of $q(x)$ multiplied by $x^{2}$.

(d) The general solution is the sum of PI and CF.