Numerical Methods in Context (Edexcel A-Level Mathematics): Flashcards

$a \frac{d^2y}{dx^2} + b \frac{dy}{dx} + cy = F(x)$

Find the complementary function (C.F.)

Assume $y = e^{mx}$ leading to $m^2 - 5m + 6 = 0$

Roots are $m = 2$ and $m = 3$ (factors: $(m-2)(m-3)=0$).

$y = C_1e^{r_1x} + C_2e^{r_2x}$ for roots $r_1, r_2$.

$y = \lambda\sin(3x) + \mu\cos(3x)$

Set coefficients of like terms equal to each other.

Use the same degree polynomial as the non-homogeneous term.

$y = y_h + y_p$

Assume $y = \lambda x e^{2x}$ if $e^{2x}$ is in C.F.