Differential equations — FAQ

A 'general solution' includes arbitrary constants (like A and B) and represents a whole family of functions that satisfy the differential equation. A 'particular solution' is found by using given initial or boundary conditions (e.g., $y=1$ when $x=0$) to determine the specific values of these constants. The particular solution is a single, unique function that satisfies both the equation and the given conditions.

Because the derivatives of sine and cosine are linked. The derivative of $\sin(ax)$ is $a\cos(ax)$, and the derivative of $\cos(ax)$ is $-a\sin(ax)$. When you substitute a trial PI into the left-hand side of the DE, which contains derivatives, you will generate both sine and cosine terms. To be able to match coefficients with the right-hand side, your trial PI must contain both terms from the outset, even if the RHS only has one of them.