The integrating factor of differential equation ( 1 - x 2 ) d y d x - x y = 1 is:

Question

The integrating factor of differential equation $(1 - x^{2}) \frac{d y}{d x} - x y = 1$ is:

Smart Achievers · Accepted Answer

(3)

Ans. $\sqrt{1 - x^{2}}$

Explanation:

Given that,

$(1 - x^{2}) \frac{d y}{d x} - x y = 1$

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

which is a linear differential equation.

$∴$ $IF = e^{\int \frac{- x}{1 - x^{2}} d x}$

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

$\Rightarrow$ $- 2 x d x = d t$

$\Rightarrow$ $x d x = - \frac{d t}{2}$

Now, $IF = e^{\frac{1}{2} \int \frac{d t}{t}} = e^{\frac{1}{2} \log t}$

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

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

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