Question

The general solution of the differential equation $\frac{dy}{dx}+\frac{2}{x}y=x^2$ is

• A. $y=c{x}^{-3}-\frac{x^2}{4}$
• B. $y=c{x}^3-\frac{x^2}{4}$
• C. $y=c{x}^2-\frac{x^3}{5}$
• D. $y=c{x}^{-2}-\frac{x^3}{5}$

Answer 1

The correct option is A $y=c{x}^{-3}-\frac{x^2}{4}$

Consider the given equation.

$\frac{dy}{dx}+\frac{2y}{x}=x^2$ $............\left(1\right)$

We know that the general equation,

$\frac{dy}{dx}+Py=Q$

Since, $P=\frac{2}{x},Q=x^2$

The integrating factor

$I.F=e^{\int Pdx}$

$I.F=e^{\int \frac{2}{x}dx}$

$I.F=e^{2lo{g}_{e}x}$

$I.F=e^{lo{g}_e{x}^2}$

$I.F=x^2$

We know that

$y\times I.F=\int I.F\times Qdx+C$

Therefore,

$x^{2}y=\int x^2\times x^{2}dx+C$

$x^{2}y=\int x^{4}dx+C$

$x^{2}y=\frac{x^5}{5}+C$

$y=\frac{x^3}{5}+x^{-2}C$