Choose the correct answer from the given four option. Integrating… - Vidyadip

MCQ

Choose the correct answer from the given four option.
Integrating factor of the differential equation $(1-\text{x}^2)\frac{\text{d}\text{y}}{\text{d}\text{x}}-\text{xy}=1$is:

A
$-\text{x}$
B
$\frac{\text{x}}{1+\text{x}^2}$
✓
$\sqrt{1-\text{x}^2}$
D
$\frac{1}{2}\log(1-\text{x}^2)$

✓

Answer

Correct option: C.

$\sqrt{1-\text{x}^2}$

Given is, $(1-\text{x}^2)\frac{\text{d}\text{y}}{\text{d}\text{x}}-\text{xy}=1$

$\Rightarrow\frac{\text{d}\text{y}}{\text{d}\text{x}}-\frac{\text{x}}{1-\text{x}^2}\text{y}=\frac{1}{1-\text{x}}^2$

Which is a linear differential equation.

$\therefore\text{I.F.}=\text{e}^{-\int\frac{\text{x}}{1-\text{x}^2}\text{dx}}$

Put $1-\text{x}^2=\text{t}$

$\Rightarrow-2\text{xdx}=\text{dt}$

$\Rightarrow\text{xdx}=-\frac{\text{dt}}{2}$

Now, $\text{I.F.}=\text{e}^{\frac{1}{2}\int\frac{\text{dt}}{\text{t}}}$

$\text{e}^{\frac{1}{2}\log\text{t}}=\text{e}^{\frac{1}{2}\log(1-\text{x}^2)}$

$\Rightarrow\sqrt{1-\text{x}^2}$

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