Solve differential equationdy+5ydx=e^(-5x)dx

First order differential equations
asked 2021-01-13

Solve differential equation \(dy+5ydx=e^{-5x}dx\)

Expert Answers (1)


\(\frac{dy}{dx}+P(x)y= Q(x)\) (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
\(I.F.= e^{\int P(x)dx}\)
\(= e^{\int 5dx}\)
\(= e^{5x}\)
\(y(I.F.)= \int Q(x)(I.F.)dx+c\)
\(\Rightarrow y(e^{5x})= \int (e^{5x})(e^{5x})dx+C\)
\(\Rightarrow y(e^{5x})= \int dx+C\)
\(= y(e^{5x})= x+C\)
\(\Rightarrow y= \frac{x+C}{e^{5x}}\)

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