\(\Rightarrow x \frac{dy}{dx}+3y= 6x^3\)

Divide by 'x'

\(\Rightarrow \frac{x \frac{dy}{dx}}{x}+\frac{3y}{x}= \frac{6x^3}{x}\)

\(\int \frac{dy}{dx}+\frac{3}{x} y= 6x^2\) (1)

Compare (1) with \(\frac{dy}{dx}+P(x)y=Q(x)\) we have

\(P(x)= \frac{3}{x}\), \(Q(x)= 6x^2\)

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

\(= e^{3 \int \frac{1}{x} dx}\)

\(= e^{3\ln(x)}\) (\(\because \int \frac{1}{x}= \ln(x)\))

\(I.F.= e^{\ln x^3}\) (\(\because n \ln x= \ln x^n\))

\(I.F.=x^3\)

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

\(\int yx^3= \int 6x^2 x^3dx\)

\(\int x^3y= 6 \int x^5dx\) (\(\because a^m a^n= a^{m+n}\))

\(\int a^n dx= \frac{a^{n+1}}{n+1}\)

\(\Rightarrow x^3y= 6 (\frac{x^{5+1}}{5+1})+C\)

\(\Rightarrow x^3y= 6 (\frac{x^6}{6})+C\)

\(\Rightarrow x^3y= x^6+C\)

\(\Rightarrow y= \frac{x^6}{x^3}+\frac{C}{x^3}\)

\(\Rightarrow y= x^3+\frac{C}{x^3}\)