Question

The general solution of the differential equation $\frac{dy}{dx}+\frac{y}{x}=3x$ is.

• A. $y=x+\frac{c}{x}$
• B. $y=x^2+\frac{c}{x}$
• C. $y=x-\frac{c}{x}$
• D. $y=x^2-\frac{c}{x^2}$

Answer 1

Answer: (B)

$y=x^2+\frac{c}{x}$

Comparing above equation with first order linear differential equation $\frac{dy}{dx}+Py=Q$

We get $P=\frac{1}{x}$ and $Q=3x$

So integration factor $I.F.=e^{\int Pdx}=e^{\int \left(1/x\right)dx}=e^{\ln x}=x$

Hence solution is given by, $y(I.F.)=\int Q(I.F.)dx$

$\Rightarrow xy=\int 3x^{2}dx$

$\Rightarrow xy=x^3+c$

$\Rightarrow y=x^2+\frac{c}{x}$