Question

I.F of $\frac{dy}{dx}+x\sin 2y=x^3{\cos }^{2}y$ is:

• A. $tany$
• B. $e^{\tan y}$
• C. $e^{s\dot{m}y}$
• D. $e^{x^2}$

Answer 1

The correct option is D $e^{x^2}$

Given the differential equation as $\frac{dy}{dx}+xsin2y=x^{3}co{s}^{2}y$

On dividing $co{s}^{2}y$ on both sides

$se{c}^{2}y\frac{dy}{dx}+x\times se{c}^{2}y\times \left(2sinycosy\right)=x^3$

$se{c}^{2}y\frac{dy}{dx}+2xtany=x^3$

Let $tany=P$

On differentiating once,

$se{c}^{2}y\frac{dy}{dx}=\frac{dP}{dx}$............(substitute the above values to given D.E)

$\Rightarrow \frac{dP}{dx}+2xP=x^3$

This is in the form of linear differential equation as $\frac{dy}{dx}+p\left(X\right)y=q\left(X\right)$.

Then the solution of the equation is $y\times u\left(x\right)=(\int (u\left(x\right)\times q\left(x\right)dx)+C$

where $u\left(x\right)=e^{\int \left(p\right(x\left)dx\right)}$ which is the integration factor of the equation.

$\Rightarrow$ On comparing the given equation with general equation.$P\left(x\right)=2x,q\left(x\right)=x^3$

$\therefore Integratingfactor=e^{\int \left(2xdx\right)}=e^{x^2}$