Question

The solution of the differential equation $\frac{dy}{dx}+xylny=x^{3}y$ is equal to (where, $C$ is the constant of integration)

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

Answer 1

Answer: (C)

$\frac{1}{y}\frac{dy}{dx}+xlny=x^3$
Let $lny=t$
$\frac{1}{y}\frac{dy}{dx}=\frac{dt}{dx}$
$\frac{dt}{dx}+tx=x^3$
I.F. $=e^{\int xdx}=e^{\frac{x^2}{2}}$
$t{e}^{\frac{x^2}{2}}=\int e^{\frac{x^2}{2}}{x}^{3}dx$
$lny{e}^{\frac{x^2}{2}}=\int \left(e^{\frac{x^2}{2}}\cdot x\right)x^{2}dx$
$lny{e}^{\frac{x^2}{2}}=e^{\frac{x^2}{2}}\cdot x^2-\int \left(2x\right)e^{\frac{x^2}{2}}dx$
$ln{e}^{\frac{x^2}{2}}=e^{\frac{x^2}{2}}{x}^2-2e^{\frac{x^2}{2}}+C$
$lny\cdot e^{\frac{x^2}{2}}=e^{\frac{x^2}{2}}\left(x^2-2\right)+C$
$lny=\left(x^2-2\right)+C{e}^{\frac{-x^2}{2}}$