The solution of differential equation \(\rm dy = \left ( 4 + y^{2} \right )dx\) is

Question

Question

Answer 1

Concept:

\(\rm \int \frac{1}{a^{2}+x^{2}}dx = \frac{1}{a}\tan ^{-1}\frac{x}{a}+ C\)

Calculation:

Given : \(\rm dy = \left ( 4 + y^{2} \right )dx\)

⇒ \(\rm \frac{dy}{4+y^{2}}= dx\)

Integrating both sides, we get

\(\rm \int \frac{dy}{2^{2}+y^{2}}= \int dx\)

⇒ \(\rm \frac{1}{2}\tan^{-1}\frac{y}{2}= x+c\)

⇒ \(\rm \tan^{-1}\frac{y}{2}= 2x+ 2c\)

⇒ \(\rm \tan^{-1}\frac{y}{2}= 2x+ C\) [∵ 2c = C]

⇒ \(\rm \frac{y}{2}= \tan(2x+ C)\)

\(\rm y = 2\tan \left ( 2x+C \right )\)

The correct option is 2 .

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