Question

Linear form of $\frac{dy}{dx}$ $+x\sin 2y$ $=$ $x^3{\cos }^{2}y$

• A. $\frac{du}{dx}+\frac{u}{x^2}=x$
• B. $\frac{du}{dx}+ux=\frac{x^3}{2}$
• C. $\frac{du}{dx}+2ux=x^3$
• D. $\frac{du}{dx}+\frac{u}{x}=x^2$

Answer 1

The correct option is C $\frac{du}{dx}+2ux=x^3$

Given , $\frac{dy}{dx}+x\sin 2y=x^3{\cos }^{2}y$

$\Rightarrow$ $\frac{dy}{dx}+2x\sin y\cos y=x^3{\cos }^{2}y$

Now dividing the equation by ${\cos }^{2}y$ we have

$\Rightarrow$ ${\sec }^{2}y\frac{dy}{dx}+2x\tan y=x^3$

Now take $\tan y=u$ $\Rightarrow$ ${\sec }^{2}y\frac{dy}{dx}=\frac{du}{dx}$

Now we have equation as :

$\Rightarrow$ $\frac{du}{dx}+2xu=x^3$ .

Hence , option 'C' is correct.