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

\(I.F.=e(\int Pdx)\)

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

\(P(x)= \frac{a}{x}\)

\(Q(x)= 40x\)

\(I.F.= e(\int \frac{a}{x} dx)= e^{a \ln x}=e^{\ln x^a}=x^a\)

\(y(I.F.)= \int(40x)(I.F.)dx + C\)

\(y(x^a)= \int(40x)(x^a)dx + C\)

\(y x^a= 40 \int x^{a+1}dx+C\)

\(y x^a= 40\times \frac{x^2+1+1}{a+1+1}+C\)

\(y x^a= 40\times\frac{x^{a+2}}{a+2}+C\)

\(y= \frac{40}{a+2}\times \frac{x^{a+2}}{x^a}+\frac{C}{x^a}\)

\(y= \frac{40}{a+2}\times x^2+\frac{C}{x^a}\) (1)

\(y(1)=a\)

Put \(x=1\) and \(y=a\) into equation (1)

\(a= \frac{40}{a+2}(1)^2+\frac{C}{1^a}\)

\(a= \frac{40}{a+2}+C\)

\(C= a-\frac{40}{a+2}\)

\(C= \frac{a(a+2)-40}{a+2}\)

\(C= \frac{a^2+2a-40}{a+2}\) Put the value of C into equation (1) \(y= \frac{40}{a+2} x^2+\frac{a^2+2a-40}{a+2}\times \frac{1}{x^a}\)