Linear Differential Equations

Let the solution curve $x=x\left(y\right),0<y<\frac{\pi }{2}$, of the differential equation ${\left({\log }_e\left(\cos y\right)\right)}^2\cos y dx-\left(1+3x{\log }_e\left(\cos y\right)\right)\sin y dy=0$  satisfy $x\left(\frac{\pi }{3}\right)=\frac{1}{2{\log }_{e}2}$. If $x\left(\frac{\pi }{6}\right)=\frac{1}{{\log }_{e}m-{\log }_{e}n}$, where $m$ and $n$ are coprime, then $mn$ is equal to

If $y=y\left(x\right)$ is the solution of the differential equation $\frac{dy}{dx}+\frac{4x}{\left(x^2-1\right)}y=\frac{x+2}{{\left(x^2-1\right)}^{\frac{5}{2}}},x>1$ such that $y\left(2\right)=\frac{2}{9}{\log }_e\left(2+\sqrt{3}\right)$ and $y\left(\sqrt{2}\right)=\alpha {\log }_e\left(\sqrt{\alpha }+\beta \right)+\beta -\sqrt{\gamma },\alpha ,\beta ,\gamma \in \mathrm{\mathbb{N}}$, then $\alpha \beta \gamma$ is equal to

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

Let$y=y\left(x\right)$ be the solution of the differential equation $\frac{dy}{dx}+\frac{5}{x\left(x^5+1\right)}y=\frac{{\left(x^5+1\right)}^2}{x^7}, x>0$. If $y\left(1\right)=2$, then $y\left(2\right)$ is equal to

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

If the solution curve of the differential equation $(y-2{\log }_{e}x)dx+\left(x{\log }_e{x}^2\right)dy=0, x>1$ passes through the points $\left(e, \frac{4}{3}\right)$ and $(e^4, \alpha )$, then $\alpha$ is equal to $\_\_\_\_\_\_\_$

Let $f$ be a differentiable function such that $x^{2}f\left(x\right)-x=4{\int }_0^{x}t f\left(t\right) dt, f\left(1\right)=\frac{2}{3}$. Then $18 f\left(3\right)$ is equal to

Let $y=y\left(x\right)$ be a solution of the differential equation $(x \cos x)dy+(xy \sin x+y \cos x-1)dx=0,0<x<\frac{\pi }{2}.$ If $\frac{\pi }{3}y\left(\frac{\pi }{3}\right)=\sqrt{3},$ then $\left|\frac{\pi }{6}{y}^{"}\left(\frac{\pi }{6}\right)+2y^{\text{'}}\left(\frac{\pi }{6}\right)\right|$ is equal to $\_\_\_\_\_\_\_.$

If the solution curve $f(x, y)=0$ of the differential equation $\left(1+{\log }_{e}x\right)\frac{dx}{dy}-x{\log }_{e}x=e^y, x>0,$ passes through the points $(1, 0)$ and $(a, 2)$, then $a^a$ is equal to

If $\left(1+x^2\right)dy=y\left(y-x\right)dx$ and $y\left(1\right)=1$ then the value of $y\left(\sqrt{2}\right)$ will be

For $\frac{dy}{dx}+\frac{5}{x\left(1+x^5\right)}y=\frac{{\displaystyle {\left(1+x^5\right)}^2}}{x^7}$; $y\left(1\right)=2$, then the value of $y\left(2\right)$ is

Solve the differential equation $\frac{dy}{dx}=-\left[\frac{ x+y\cos x}{1+\sin x}. \right]$

The solution of $\frac{dy}{dx}=\frac{x^2-y^2+1}{2xy}$

Let $f:R\to R$ be a continuous function satisfying $f\left(x\right)=e^{\raisebox{1ex}{$x^2$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+{\int }_0^{x}tf\left(t\right)dt$ for all $x$. Then, which of the following is correct?

Let $y=y\left(x\right)$ be the solution curve of the differential equation $\left(y^2-x\right)\left(\frac{dy}{dx}\right)=1$ satisfying $y\left(0\right)=1$. This curve intersects $x-\mathrm{axis}$ at the point whose abscissa is: 

If $y\left(x\right)$ be the solution of differential equation $x\log x\frac{dy}{dx}+y=2x\log x,$ then $y\left(\mathrm{e}\right)$ is equal to

Let $y=y\left(x\right)$ be the solution of the differential equation $\left(1-x^2\right)dy=\left(xy+\left(x^3+2\right)\sqrt{1-x^2}\right)dx,-1<x<1$. and $y\left(0\right)=0$. If ${\int }_{\frac{-1}{2}}^{\frac{1}{2}}\sqrt{1-x^2}y\left(x\right)dx=k$, then $k^{-1}$ is equal to

Solution of the differential equation $\left(1+y^2\right)dx=\left({\tan }^{-1}y-x\right)dy$ is
(where $c$ is an arbitrary constant)

Consider the differential equation, $ydx-\left(x+y^2\right)dy=0$. If for $y=1, x$ takes value $1$, then value of $x$ when $y=4$ is

Let $y=y\left(x\right)$ be the solution of the differential equation $x{\log }_{e}x\frac{dy}{dx}+y=3x{\log }_{e}x, \left(x>1\right).$ If $y\left(e\right)=0,$ then $y\left(e^2\right)$ is equal to