Question

The function $y$ specified implicitly by the relation ${\int }_0^y{e}^{t}dt+{\int }_0^x\cos tdt=0$ satisfies the differential equation

• A. $e^{2y}(\frac{d^{2}y}{d{x}^2}+{\left(\frac{dy}{dx}\right)}^2)=\sin x$
• B. $e^y(\frac{d^{2}y}{d{x}^2}+{\left(\frac{dy}{dx}\right)}^2)=\sin 2x$
• C. $e^y(2\frac{d^{2}y}{d{x}^2}+{\left(\frac{dy}{dx}\right)}^2)=\sin x$
• D. $e^y(\frac{d^{2}y}{d{x}^2}+{\left(\frac{dy}{dx}\right)}^2)=\sin x$

Answer 1

Answer: (D)

$e^y(\frac{d^{2}y}{d{x}^2}+{\left(\frac{dy}{dx}\right)}^2)=\sin x$

Given, ${\int }_0^y{e}^{t}dt+{\int }_0^x\cos tdt=0$

$\Rightarrow {\left[e^t\right]}_0^y+{\left[\sin x\right]}_0^x=0$

$\Rightarrow e^y+\sin x=0$

By differentiating w.r.t $x$, we get

$e^y\frac{dy}{dx}+\cos x=0$

Again, by differentiating w.r.t $x$ we get

$e^y.\frac{dy}{dx}.\frac{dy}{dx}+e^y\frac{d^{2}y}{d{x}^2}-\sin x=0$

$\Rightarrow e^y(\frac{d^{2}y}{d{x}^2}+{\left(\frac{dy}{dx}\right)}^2)=\sin x$