Suppose for a differentiable function $$h,h(0)=0,h(1)=1$$ and $$h^{\prime }(0)=h^{\prime }(1)=2$$. If $$g(x)=h\left({\mathrm{e}}^x\right){\mathrm{e}}^{h(x)}$$, then $$g^{\prime }(0)$$ is equal to:

Let $$f:(-\infty ,\infty )-\{0\}\to \mathbb{R}$$ be a differentiable function such that f^{\prime}(1)=\lim _\limits{a \rightarrow \infty} a^2 f\left(\frac{1}{a}\right). Then \lim _\limits{a \rightarrow \infty} \frac{a(a+1)}{2} \tan ^{-1}\left(\frac{1}{a}\right)+a^2-2 \log _e a is equal to

Let $\alpha$ be a non-zero real number. Suppose $f:\mathbf{R}\to \mathbf{R}$ is a differentiable function such that $f(0)=2$ and $\underset{x\to -\infty }{\lim }f(x)=1$. If $f^{\prime }(x)=\alpha f(x)+3$, for all $x\in \mathbf{R}$, then $f\left(-{\log }_{\mathrm{e}}2\right)$ is equal to :

Let $y=y(x)$ be the solution of the differential equation $\frac{\mathrm{d}y}{\mathrm{d}x}=2x(x+y{)}^3-x(x+y)-1,y(0)=1$. Then, ${\left(\frac{1}{\sqrt{2}}+y\left(\frac{1}{\sqrt{2}}\right)\right)}^2$ equals :

Let $x=x(\mathrm{t})$ and $y=y(\mathrm{t})$ be solutions of the differential equations $\frac{\mathrm{d}x}{\mathrm{dt}}+\mathrm{a}x=0$ and $\frac{\mathrm{d}y}{\mathrm{dt}}+\mathrm{by}=0$ respectively, $\mathrm{a},\mathrm{b}\in \mathbf{R}$. Given that $x(0)=2 ; y(0)=1$ and $3 y(1)=2 x(1)$, the value of $\mathrm{t}$, for which $x(\mathrm{t})=y(\mathrm{t})$, is :

If $$y=y(x)$$ is the solution curve of the differential equation $$\left(x^2-4\right)\mathrm{d}y-\left(y^2-3y\right)\mathrm{d}x=0,x>2,y(4)=\frac{3}{2}$$ and the slope of the curve is never zero, then the value of $$y(10)$$ equals :

The temperature $$T(t)$$ of a body at time $$t=0$$ is $$160^{\circ }\mathrm{F}$$ and it decreases continuously as per the differential equation $$\frac{dT}{dt}=-K(T-80)$$, where $$K$$ is a positive constant. If $$T(15)=120^{\circ }\mathrm{F}$$, then $$T(45)$$ is equal to

Let $$y=y(x)$$ be the solution of the differential equation $$\frac{dy}{dx}=\frac{(\tan x)+y}{\sin x(\sec x-\sin x\tan x)},x\in \left(0,\frac{\pi }{2}\right)$$ satisfying the condition $$y\left(\frac{\pi }{4}\right)=2$$. Then, $$y\left(\frac{\pi }{3}\right)$$ is

The solution curve of the differential equation $$y\frac{dx}{dy}=x\left({\log }_{e}x-{\log }_{e}y+1\right),x>0,y>0$$ passing through the point $$(e,1)$$ is

A function $$y=f(x)$$ satisfies $$f(x)\sin 2x+\sin x-\left(1+{\cos }^{2}x\right)f^{\prime }(x)=0$$ with condition $$f(0)=0$$. Then, $$f\left(\frac{\pi }{2}\right)$$ is equal to

If $$\sin \left(\frac{y}{x}\right)={\log }_e|x|+\frac{\alpha }{2}$$ is the solution of the differential equation $$x\cos \left(\frac{y}{x}\right)\frac{dy}{dx}=y\cos \left(\frac{y}{x}\right)+x$$ and $$y(1)=\frac{\pi }{3}$$, then $${\alpha }^2$$ is equal to

Let $$y=y(x)$$ be the solution of the differential equation $$\sec x\mathrm{d}y+\{2(1-x)\tan x+x(2-x)\}\mathrm{d}x=0$$ such that $$y(0)=2$$. Then $$y(2)$$ is equal to:

Let \int_\limits0^x \sqrt{1-\left(y^{\prime}(t)\right)^2} d t=\int_0^x y(t) d t, 0 \leq x \leq 3, y \geq 0, y(0)=0. Then at $$x=2,y^{\prime \prime }+y+1$$ is equal to

The solution of the differential equation $$(x^2+y^2)\mathrm{d}x-5xy\mathrm{d}y=0,y(1)=0$$, is :

The solution curve, of the differential equation $$2y\frac{\mathrm{d}y}{\mathrm{d}x}+3=5\frac{\mathrm{d}y}{\mathrm{d}x}$$, passing through the point $$(0,1)$$ is a conic, whose vertex lies on the line :

If the solution $$y=y(x)$$ of the differential equation $$(x^4+2x^3+3x^2+2x+2)\mathrm{d}y-(2x^2+2x+3)\mathrm{d}x=0$$ satisfies $$y(-1)=-\frac{\pi }{4}$$, then $$y(0)$$ is equal to :

Let $$y=y(x)$$ be the solution of the differential equation $$(x^2+4{)}^{2}dy+(2x^{3}y+8xy-2)dx=0$$. If $$y(0)=0$$, then $$y(2)$$ is equal to

Let $$y=y(x)$$ be the solution curve of the differential equation $$\sec y\frac{\mathrm{d}y}{\mathrm{d}x}+2x\sin y=x^3\cos y,y(1)=0$$. Then $$y(\sqrt{3})$$ is equal to:

Let $$f(x)$$ be a positive function such that the area bounded by $$y=f(x),y=0$$ from $$x=0$$ to $$x=a>0$$ is $$e^{-a}+4a^2+a-1$$. Then the differential equation, whose general solution is $$y=c_{1}f(x)+c_2$$, where $$c_1$$ and $$c_2$$ are arbitrary constants, is