Differential Equations

Let $y = y(x)$ be the solution of the differential equation ${x^3}dy + (xy - 1)dx = 0,x > 0,y\left( {{1 \over 2}} \right) = 3 - \mathrm{e}$. Then y (1) is equal to

Let $x=x(y)$ be the solution of the differential equation $2(y+2) \log _{e}(y+2) d x+\left(x+4-2 \log _{e}(y+2)\right) d y=0, y>-1$ with $x\left(e^{4}-2\right)=1$. Then $x\left(e^{9}-2\right)$ is equal to :

Let $y=y(x), y > 0$, be a solution curve of the differential equation $\left(1+x^{2}\right) \mathrm{d} y=y(x-y) \mathrm{d} x$. If $y(0)=1$ and $y(2 \sqrt{2})=\beta$, then

Let $f$ be a differentiable function such that ${x^2}f(x) - x = 4\int\limits_0^x {tf(t)dt}$, $f(1) = {2 \over 3}$. Then $18f(3)$ is equal to :

Let $\alpha$ be a non-zero real number. Suppose $f: \mathbf{R} \rightarrow \mathbf{R}$ is a differentiable function such that $f(0)=2$ and $\lim\limits_{x \rightarrow-\infty} f(x)=1$. 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 $x=x(\mathrm{t})$ 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 :

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 $\frac{d T}{d t}=-K(T-80)$, where $K$ is a positive constant. If $T(15)=120^{\circ} \mathrm{F}$, then $T(45)$ is equal to

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

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 $(x^2+y^2) \mathrm{d} x-5 x y \mathrm{~d} y=0, y(1)=0$, is :