Question

The Integrating Factor of the differential equation $x\frac{dy}{dx}-y=2x^2$ is
(A) $e^{-x}$
(B) $e^{-y}$
(C) $\frac{1}{x}$
(D) x

Answer 1

Answer:

option (c) is correct

Step-by-step explanation:

Concept:

The integrating factor of the differential equation

$\frac{dy}{dx}+Py=Q\:is\:e^{\int{P}\:dx}$

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

Divide both sides by x

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

comparing this equation with

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

we get

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

Integrating factor

$=e^{\int{P}\:dx}\\\\=e^{\int{\frac{-1}{x}}\:dx}\\\\=e^{-\int{\frac{1}{x}}\:dx}\\\\=e^{-logx}\\\\=e^{logx^{-1}}\\\\=x^{-1}$

$=\frac{1}{x}$

Note:

$e^{logA}=A$