Question

The integrating factor of the differential equation $x\frac{dy}{dx}+y=x^3{y}^6$ is

• A. $-\frac{5}{x^5}$
• B. $\frac{5}{y^5}$
• C. $-\frac{1}{x^5}$
• D. $\frac{1}{x^5}$

Answer 1

Answer: (D)

$\frac{1}{x^5}$

$x\frac{dy}{dx}+y=x^3{y}^6$
$\Rightarrow$ $\frac{1}{y^6}\frac{dy}{dx}+\frac{1}{y^5}\frac{1}{x}=x^2$ (*)
Substitute $\frac{1}{y^5}=t$ $\therefore$ $\frac{1}{y^6}\frac{dy}{dx}=-\frac{1}{5}\frac{dt}{dx}$
$\therefore$ $\frac{dt}{dx}-\frac{5}{x}t=-5x^2$ which is linear in t
$\therefore$ I.F. $=e^{\int \frac{-5}{x}dx}=\frac{1}{x^5}$