Let $x=x(t)$ and $y=y(t)$ be solutions of the differential equations $\frac{\mathrm{dx}}{\mathrm{dt}}+\mathrm{ax}=0$ and $\frac{\mathrm{dy}}{\mathrm{dt}}+\mathrm{by}=0$ respectively, $\mathrm{a}, \mathrm{b} \in \mathrm{R}$. Given that $x(0)=2 ; y(0)=1$ and $...

If for the solution curve $y=f(x)$ of the differential equation $\frac{\mathrm{d} y}{\mathrm{~d} x}+(\tan x) y=\frac{2+\sec x}{(1+2 \sec x)^{2}}$, $x \in\left(\frac{-\pi}{2}, \frac{\pi}{2}\right), f\left(\frac{\pi}{3}\right)=\frac{\sqrt{3}}{10}$, then $f\left(\frac{\pi}{4}\right)$ is...

If $y=y(x)$ is the solution of the differential equation, $\sqrt{4-x^{2}} \frac{\mathrm{~d} y}{\mathrm{~d} x}=\left(\left(\sin ^{-1}\left(\frac{x}{2}\right)\right)^{2}-y\right) \sin ^{-1}\left(\frac{x}{2}\right),-2 \leq x \leq 2, y(2)=\frac{\pi^{2}-8}{4}$, then $y^{2}(0)$ is equal...

Let $y = y(x)$ be the solution curve of the differential equation $...

Let $f:[0,\infty)\to$ $\mathbb{R}$ be a differentiable function such that $...

Let g be a differentiable function such that $\int_0^x g (t)dt = x - \int_0^x t g(t)dt,x \ge 0$ and let $y = y(x)$ satisfy...

Let $f:R \to R$ be a thrice differentiable odd function satisfying ${f^\prime }(x) \ge 0,{f^{\prime \prime }}(x) = f(x),f(0) = 0,{f^\prime }(0) = 3$. Then...

Let $y=y(x)$ be the solution of the differential equation $2 \cos x \frac{\mathrm{~d} y}{\mathrm{~d} x}=\sin 2 x-4 y \sin x, x \in\left(0, \frac{\pi}{2}\right)$. If $y\left(\frac{\pi}{3}\right)=0$,...