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. $y=x^2-\frac{c}{x}$

Answer 1

Answer: (B), (D)

Given differential equation is $\frac{dy}{dx}+\frac{y}{x}=3x$
It is a linear differential equation of the form
$\frac{dY}{dx}+Py=Q$
$\therefore P=\frac{1}{x}\text{ and }Q=3x$
$\therefore IF=e^{\int Pdx}=e^{\int \frac{1}{x}dx}$
$=e^{\log x}=x$
$\therefore$ Complete solution is
$yx=\int 3x\times xdx+C$...(i)
$\Rightarrow yx=3\left[\frac{x^3}{3}\right]+C$
$\Rightarrow y=x^2+\frac{C}{x}$
Also, Eq. (i) can be written as
$yx=\int 3x\times xdx-C$
$\Rightarrow yx=x^3-C$
$\Rightarrow y=x^2-\frac{C}{x}$