If $y=y(x)$ is the solution of the differential equation $\frac{5+\mathrm{e}^{\mathrm{x}}}{2+\mathrm{y}} \cdot \frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{e}^{\mathrm{x}}=0$ satisfying $\mathrm{y}(0)=1$, then a value of $y\left(\log _e 13\right)$ is :

The general solution of the differential equation $\sqrt{1+x^2+y^2+x^2 y^2}+x y \frac{d y}{d x}=0$ is : (where C is a constant of integration)

If the lines $x+y=a$ and $x-y=b$ touch the curve $y=x^2-3 x+2$ at the points where the curve intersects the $x$-axis, then $\frac{a}{b}$ is equal to

Let the normal at a point $P$ on the curve $y^2-3 x^2+y+10=0$ intersect the $y$-axis at $\left(0, \frac{3}{2}\right)$. If $m$ is the slope of the...

Let $y=y(x)$ be the solution of the differential equation, $\frac{d y}{d x}+y \tan 2 x=x^2+x^2 \tan x$, $x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, such that $y(0)=1$.Then :

Let $y=y(x)$ be the solution of the differential equation $\cos x \frac{d y}{d x}+2 y \sin x=\sin 2 x, x \in\left(0, \frac{\pi}{2}\right)$. If $...

If the solution curve of the differential equation $\left(2 x-10 y^3\right) d y+y d x=0$, passes through the points $(0,1)$ and $(2, \beta)$, then $b$...

A differential equation representing the family of parabolas with axis parallel to y-axis and whose length of latus rectum is the distance of the point...

If $\frac{d y}{d x}+\frac{3}{\cos ^2 x} y=\frac{1}{\cos ^2 x}, x \in\left(\frac{-\pi}{3}, \frac{\pi}{3}\right)$ and $y\left(\frac{\pi}{4}\right)=\frac{4}{3}$, they $y\left(-\frac{\pi}{4}\right)$ equals