Question

An integrating factor of the differential equation $x\frac{dy}{dx}-y=x^3;x>0$ is ______

• A. $x$
• B. $-\frac{1}{x}$
• C. $-x$
• D. $\frac{1}{x}$

Answer 1

Answer: (D)

$\frac{1}{x}$

For $\frac{dy}{dx}+P\left(x\right)y=Q\left(x\right)$ where &&P(x),Q(x)$ are function of x

The integrating factor$=e^{\int P\left(x\right)dx}$

Here $x\frac{dy}{dx}-y=x^3$

$\frac{dy}{dx}-\frac{y}{x}=x^2$

$P\left(x\right)=-\frac{1}{x}$

Integrating Factor $=e^{\int -\frac{1}{x}dx}=e^{-lnx}=e^{ln{x}^{-1}}=e^{ln\frac{1}{x}}=\frac{1}{x}$