Question

Solve: $\frac{dy}{dx}+2y=\sin x$

Answer 1

We have,

$\frac{dy}{dx}+2y=\sin x$ ……. (1)

Since, $P=2,Q=\sin x$

Integrating factor,

$I.F=e^{\int Pdx}$

$I.F=e^{\int 2dx}$

$I.F=e^{2x}$

Therefore, the general solution

$y\times I.F=\int Q\times I.Fdx$

$y\times e^{2x}=\int \sin x\times e^{2x}dx$

We know that

$\int e^{ax}\sin bxdx=\frac{e^{ax}}{a^2+b^2}(a\sin bx-b\cos bx)+C$

Thus,

$y\times e^{2x}=\frac{e^{2x}}{2^2+1^2}(2\sin x-\cos x)+C$

$y\times e^{2x}=\frac{e^{2x}}{5}(2\sin x-\cos x)+C$

Hence, this is the answer.