Finding the PS (of a DE)

For an equation of the form:
$\mathsf{\frac{dy}{dx}+P(x)\cdot y = Q(x)}$ (in this case $\mathsf{P(x)=6}$ and $\mathsf{Q(x)=5\sin{2x}}$ )

$\mathsf{y=\frac{\int Q(x)\cdot e^{\int P(x)dx}dx}{e^{\int P(x)dx}}}$

For this case:

$\mathsf{\int P(x)dx=6x}$ and hence $\mathsf{e^{\int P(x)dx}=e^{6x}}$

So to the guts of the problem (remembering the constant of integration at this stage):

$\mathsf{y=\frac{\int 5\sin{2x}\cdot e^{6x}dx}{e^{6x}}=\frac{e^{6x}\cdot \frac{3}{4}\sin{2x}-e^{6x}\cdot \frac{1}{4}\cos{2x}+C}{e^{6x}}=\frac{3}{4}\sin{2x}-\frac{1}{4}\cos{2x}+Ce^{-6x}}$

$\mathsf{\frac{dy}{dx}=\frac{3}{2}\cos{2x}+\frac{1} {2}\sin{2x}-6Ce^{-6x}}$

Checking:

$\mathsf{\frac{dy}{dx}+6y=5\sin{2x}}$

$\mathsf{\frac{3}{2}\cos{2x}+\frac{1}{2}\sin{2x}-6Ce^{-6x}+6(\frac{3}{4}\sin{2x}-\frac{1}{4}\cos{2x}+Ce^{-6x})=5\sin{2x}}$

To determine C you'll need a known point