If e^y + xy = e,then the value of frac{d^2y}{ | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Given the equation $e^y + xy = e$. Differentiating with respect to $x$,we get: $e^y \frac{dy}{dx} + y + x \frac{dy}{dx} = 0$ .....$(i)$ Differenti

Vedclass · Accepted Answer

$\frac{1}{e^2}$

Vedclass · Answer

(B) Given the equation $e^y + xy = e$. Differentiating with respect to $x$,we get: $e^y \frac{dy}{dx} + y + x \frac{dy}{dx} = 0$ .....$(i)$ Differentiating again with respect to $x$,we get: $e^y \frac{d^2y}{dx^2} + e^y \left( \frac{dy}{dx} \right)^2 + \frac{dy}{dx} + \frac{dy}{dx} + x \frac{d^2y}{dx^2} = 0$ $e^y \frac{d^2y}{dx^2} + e^y \left( \frac{dy}{dx} \right)^2 + 2 \frac{dy}{dx} + x \frac{d^2y}{dx^2} = 0$ .....$(ii)$ Putting $x = 0$ in the original equation $e^y + xy = e$,we get $e^y + 0 = e$,which implies $y = 1$. Substituting $x = 0$ and $y = 1$ in equation $(i)$: $e^1 \frac{dy}{dx} + 1 + 0 = 0$ $e \frac{dy}{dx} = -1$ $\frac{dy}{dx} = -\frac{1}{e}$ Substituting $x = 0$,$y = 1$,and $\frac{dy}{dx} = -\frac{1}{e}$ in equation $(ii)$: $e \frac{d^2y}{dx^2} + e \left( -\frac{1}{e} \right)^2 + 2 \left( -\frac{1}{e} \right) + 0 = 0$ $e \frac{d^2y}{dx^2} + e \left( \frac{1}{e^2} \right) - \frac{2}{e} = 0$ $e \frac{d^2y}{dx^2} + \frac{1}{e} - \frac{2}{e} = 0$ $e \frac{d^2y}{dx^2} - \frac{1}{e} = 0$ $e \frac{d^2y}{dx^2} = \frac{1}{e}$ $\frac{d^2y}{dx^2} = \frac{1}{e^2}$

If $e^y + xy = e$,then the value of $\frac{d^2y}{dx^2}$ for $x = 0$ is

Similar Questions

If $f(x) = (x^2 - 1)^7$, then $f^{(14)}(x)$ is equal to

If $f(x)=b \cdot e^{a x}+a \cdot e^{b x}$,then $f^{\prime \prime}(0)=$

If $f(x) = \sin x + \cos x$, then $f\left(\frac{\pi}{4}\right) f^{(iv)}\left(\frac{\pi}{4}\right)$ is equal to

If $\sqrt{1-x^2}+\sqrt{1-y^2}=a(x-y)$, then $\left[\left(1-x^2\right)^2 \frac{d^2 y}{d x^2}+y\left(1-x^2\right)\right] \frac{d y}{d x}=$

If $y = \sqrt{x^2 + 8x + 3}$, then the value of $\frac{d^2y}{dx^2}$ at $x = 3$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module