Question

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

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

Answer 1

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

Using Newton Leibnitz rule, we get
$e^y\frac{dy}{dx}+\cos x=0\Rightarrow e^y\frac{d^{2}y}{d{x}^2}+e^y{\left(\frac{dy}{dx}\right)}^2-\sin x=0$
$\Rightarrow e^y(\frac{d^{2}y}{d{x}^2}+{\left(\frac{dy}{dx}\right)}^2)=\sin x$