\(dy/dx+P(x)y= Q(x)\) (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

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

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

\(= e^(5x)\)

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

\(=> y(e^5x)= int (e^5x)(e^5x)dx+C\)

\(=> y(e^5x)= int dx+C\)

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

\(=> y= (x+C)/e^(5x)\)

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

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

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

\(= e^(5x)\)

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

\(=> y(e^5x)= int (e^5x)(e^5x)dx+C\)

\(=> y(e^5x)= int dx+C\)

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

\(=> y= (x+C)/e^(5x)\)