Differential equations — common mistakes

A very common mistake is forgetting to check if your chosen form for the Particular Integral is already part of the Complementary Function. If $f(x) = 3e^{2x}$ and your CF is $y_{CF} = Ae^{2x} + Be^{x}$, you cannot use $Ce^{2x}$ for the PI. You must multiply by $x$ and try $y_{PI} = Cxe^{2x}$. If the root in the CF was repeated, you would multiply by $x^2$.

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.