\(\frac{dy}{dx}+P(x)y= Q(x)\) (1)

where P(x) or Q(x) are constants or function of x alone

Integrating factor of (1) is

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

Required solution is

\(y (I.F.)= \int Q(x)(I.F)dx+C\)

\(dy+(5y)dx= e^{-5x}dx\)

\(\Rightarrow \frac{dy}{dx}+5y= e^{-5x} (*)\)

Since this equation is in the form \(\frac{dy}{dx}+P(x)y=Q(x)\) so it is linear ODE comparing \((*)\) with \(\frac{dy}{dx}+P(x)y= Q(x)\) we get \(P(x)=5\), \(Q(x)= e^{−5x}\)

So integrating factor is

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

\(= e^{\int 5dx}\)

\(= e^{5x}\)

\(y(I.F.)= \int Q(x)(I.F.)dx+c\)

\(\Rightarrow y(e^{5x})= \int (e^{5x})(e^{5x})dx+C\)

\(\Rightarrow y(e^{5x})= \int dx+C\)

\(= y(e^{5x})= x+C\)

\(\Rightarrow y= \frac{x+C}{e^{5x}}\)