Question

The Integrating Factor of the differential equation ${\displaystyle \frac{dy}{dx}-y=2x^2}$ is ______.

विकल्प

• e^-x

• e^-y

• ${\displaystyle \frac{1}{x}}$

• x

Answer 1

The Integrating Factor of the differential equation ${\displaystyle \frac{dy}{dx}-y=2x^2}$ is ${\displaystyle \underline{\frac{1}{x}}}$

Explanation:

The differential equation is

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

or ${\displaystyle \frac{dy}{dx}-\frac{1}{x}y=2x}$

Here ${\displaystyle P=-\frac{1}{x},Q=2x}$

∴ ${\displaystyle \int P dx=\int -\frac{1}{x}}$ dx

=${\displaystyle -\log x=\log \frac{1}{x}}$

⇒ ${\displaystyle I.F.=e^{\int Pdx}}$

${\displaystyle =e^{\log 1/x}=\frac{1}{x}}$