Differential Equations

For $x \in \mathbb{R}$, let the function $y(x)$ be the solution of the differential equation
$$\frac{dy}{dx} + 12y = \cos\left(\frac{\pi}{12}x\right), \qquad y(0) = 0.$$
Then, which of the following statements is/are TRUE?

If $x=x(y)$ is the solution of the differential equation $y\dfrac{dx}{dy}=2x+y^3(y+1)e^y$, $x(1)=0$; then $x(e)$ is equal to :

A particle is moving in the $xy$-plane along a curve $C$ passing through the point $(3,3)$. The tangent to the curve $C$ at the point $P$ meets the $x$-axis at $Q$. If the $y$-axis bisects the segment $PQ$, then $C$ is a parabola with

The slope of normal at any point $(x,y)$, $x>0$, $y>0$ on the curve $y=y(x)$ is given by $\dfrac{x^2}{xy-x^2y^2-1}$. If the curve passes through the point $(1,1)$, then $e\cdot y(e)$ is equal to

Let $f(x)$ be a polynomial function such that $f(x)+f'(x)+f''(x)=x^5+64$. Then, the value of $\displaystyle\lim_{x\to1}\frac{f(x)}{x-1}$ is equal to :

Let $y=y(x)$ be the solution of the differential equation $(x+1)y'-y=e^{3x}(x+1)^2$, with $y(0)=\dfrac13$. Then, the point $x=-\dfrac43$ for the curve $y=y(x)$ is :

If the solution curve $y=y(x)$ of the differential equation $y^2dx+\left(x^2-xy+y^2\right)dy=0$, which passes through the point $(1,1)$ and intersects the line $y=\sqrt3x$ at the point $\left(\alpha,\sqrt3\alpha\right)$, then value of $\log_e\left(\sqrt3\alpha\right)$ is equal to :

If $y=y(x)$ is the solution of the differential equation $2x^2\dfrac{dy}{dx}-2xy+3y^2=0$ such that $y(e)=\dfrac e3$, then $y(1)$ is equal to

Let the solution curve $y=y(x)$ of the differential equation $\left(4+x^2\right)dy-2x\left(x^2+3y+4\right)dx=0$ pass through the origin. Then $y(2)$ is equal to ______.

Let $S=(0,2\pi)-\left\{\dfrac\pi2,\dfrac{3\pi}4,\dfrac{3\pi}2,\dfrac{7\pi}4\right\}$. Let $y=y(x)$, $x\in S$, be the solution curve of the differential equation $\dfrac{dy}{dx}=\dfrac1{1+\sin2x}$, $y\left(\dfrac\pi4\right)=\dfrac12$. If the sum of abscissas of all the points of intersection of the curve $y=y(x)$ with the curve $y=\sqrt2\sin x$ is $\dfrac{k\pi}{12}$, then $k$ is equal to ______.

If $y=y(x)$ is the solution of the differential equation $x\dfrac{dy}{dx}+2y=xe^x$, $y(1)=0$ then the local maximum value of the function $z(x)=x^2y(x)-e^x$, $x\in\mathbb R$ is :

If the solution of the differential equation
$$\frac{dy}{dx}+e^x\left(x^2-2\right)y=\left(x^2-2x\right)\left(x^2-2\right)e^{2x}$$
satisfies $y(0)=0$, then the value of $y(2)$ is ______

Let $\dfrac{dy}{dx}=\dfrac{ax-by+a}{bx+cy+a}$, where $a,b,c$ are constants, represent a circle passing through the point $(2,5)$. Then the shortest distance of the point $(11,6)$ from this circle is :

If $\dfrac{dy}{dx}+\dfrac{2^{x-y}\left(2^y-1\right)}{2^x-1}=0$, $x,y>0$, $y(1)=1$, then $y(2)$ is equal to :

If the solution curve of the differential equation $\left(\left(\tan^{-1}y\right)-x\right)dy=\left(1+y^2\right)dx$ passes through the point $(1,0)$, then the abscissa of the point on the curve whose ordinate is $\tan(1)$, is

Let $y=y(x)$ 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 $y(0)=0$. If $\displaystyle\int_{-1/2}^{1/2}\sqrt{1-x^2}\,y(x)\,dx=k$, then $k^{-1}$ is equal to ______.

Let the solution curve $y=y(x)$ of the differential equation
$$\left[\frac x{\sqrt{x^2-y^2}}+e^{\frac yx}\right]x\frac{dy}{dx}=x+\left[\frac x{\sqrt{x^2-y^2}}+e^{\frac yx}\right]y$$
pass through the points $(1,0)$ and $(2\alpha,\alpha)$, $\alpha>0$. Then $\alpha$ is equal to

Let $y=y(x)$ be the solution of the differential equation $x\left(1-x^2\right)\dfrac{dy}{dx}+\left(3x^2y-y-4x^3\right)=0$, $x>1$, with $y(2)=-2$. Then $y(3)$ is equal to

Let $x=x(y)$ be the solution of the differential equation $2y\,e^{x/y^2}dx+\left(y^2-4xe^{x/y^2}\right)dy=0$ such that $x(1)=0$. Then, $x(e)$ is equal to :

Let the slope of the tangent to a curve $y=f(x)$ at $(x,y)$ be given by $2\tan x(\cos x-y)$. If the curve passes through the point $\left(\frac\pi4,0\right)$, then the value of $\displaystyle\int_0^{\pi/2}y\,dx$ is equal to :