Question

Find the general solution of $\frac{dy}{dx}+y=e^x$.

Answer 1

Given the differential equation is

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

It is of type

$\frac{dy}{dx}+P\left(x\right)y=Q\left(x\right)$

I.F. = $e^{\int P\left(x\right).dx}$

Now the integrating factor for the differential equation will be $e^{\int 1.dx}=e^x$

Now multiplying both sides of the given differential equation by $e^x$ and then integrating we've,

$y{e}^x=\int e^{2x}dx+c$

or, $y{e}^x=\frac{e^{2x}}{2}+c$

or, $y=\frac{e^x}{2}+c{e}^{-x}$. [ Where $c$ is integrating constant]