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

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

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

Answers (1)

2021-01-14
\(dy/dx+P(x)y= Q(x)\) (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
\(y (I.F.)= int Q(x)(I.F)dx+C\)
\(dy+(5y)dx= e^(-5x)dx\)
\(=> dy/dx+5y= e^(-5x)\( (*)
Since this equation is in the form \(dydx+P(x)y=Q(x)\) so it is linear ODE comparing (*) with \(dy/dx+P(x)y= Q(x)\) we get P(x)=5, \(Q(x)= e^(−5x)\)
So integrating factor is
\(I.F.= e^(int P(x)dx)\)
\(= e^(int 5dx)\)
\(= e^(5x)\)
\(y(I.F.)= int Q(x)(I.F.)dx+c\)
\(=> y(e^5x)= int (e^5x)(e^5x)dx+C\)
\(=> y(e^5x)= int dx+C\)
\(= y(e^5x)= x+C\)
\(=> y= (x+C)/e^(5x)\)
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