Question

Find the general solution of the differential equation $\frac{dy}{dx}+3y=e^{-2x}$

Answer 1

Given differential equation :

$\frac{dy}{dx}+3y=e^{-2x}$

Given differential equation is of the form

$\frac{dy}{dx}+Py=Q$

By comparing both the equations, we get

$P=3$ and $Q=e^{-2x}$

The general solution of the given differential equation is

$y(I.F)=\int (Q\times I.F.)dx+c$ ....(i)

Firstly, we need to find I.F.

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

$I.F.=e^{\int 3dx}$

$I.F.=e^{3x}$

Substituting the value of I.F in (i), we get

$y\times 2^{3x}=\int e^{-2x}.e^{3x}dx+c$

$y{e}^{3x}=\int e^{x}dx+c$

$y{e}^{3x}=e^x+c$

Hence, the required general solution is
$y=e^{-2x}+c{e}^{-3x}$