Differential Equations

Let the solution curve of the differential equation $x\dfrac{dy}{dx}-y=\sqrt{y^2+16x^2}$, $y(1)=3$ be $y=y(x)$. Then $y(2)$ is equal to :

Let $y=y(x)$ be the solution of the differential equation
$$\frac{dy}{dx}+\frac{\sqrt2y}{2\cos^4x-\cos2x}=xe^{\tan^{-1}\left(\sqrt2\cot2x\right)},\quad0<x<\frac\pi2$$
with $y\left(\dfrac\pi4\right)=\dfrac{\pi^2}{32}$. If $y\left(\dfrac\pi3\right)=\dfrac{\pi^2}{18}e^{-\tan^{-1}(\alpha)}$, then the value of $3\alpha^2$ is equal to ______.

If $y=y(x)$ is the solution of the differential equation $\left(1+e^{2x}\right)\dfrac{dy}{dx}+2\left(1+y^2\right)e^x=0$ and $y(0)=0$, then $6\left(y'(0)+\left(y\left(\log_e\sqrt3\right)\right)^2\right)$ is equal to

Let $y=y(x)$, $x>1$, be the solution of the differential equation $(x-1)\dfrac{dy}{dx}+2xy=\dfrac1{x-1}$, with $y(2)=\dfrac{1+e^4}{2e^4}$. If $y(3)=\dfrac{e^\alpha+1}{\beta e^\alpha}$, then the value of $\alpha+\beta$ is equal to ______.

Let $\dfrac{dy}{dx}=\dfrac{ax-by+a}{bx+cy+a}$, $a,b,c\in\mathbf R$, represents a circle with center $(\alpha,\beta)$. Then, $\alpha+2\beta$ is equal to :

A hostel has 100 students. On a certain day (consider it day zero) it was found that two students are infected with some virus. Assume that the rate at which the virus spreads is directly proportional to the product of the number of infected students and the number of non-infected students. If the number of infected students on 4$^{\text{th}}$ day is 30, then number of infected students on 8$^{\text{th}}$ day will be ______.

The slope of the tangent to a curve $C:y=y(x)$ at any point $(x,y)$ on it is $\dfrac{2e^{2x}-6e^{-x}+9}{2+9e^{-2x}}$. If C passes through the points $\left(0,\dfrac12+\dfrac\pi{2\sqrt2}\right)$ and $\left(\alpha,\dfrac12e^{2\alpha}\right)$, then $e^\alpha$ is equal to :

The general solution of the differential equation $\left(x-y^2\right)dx+y\left(5x+y^2\right)dy=0$ is :

Let a smooth curve $y=f(x)$ be such that the slope of the tangent at any point $(x,y)$ on it is directly proportional to $\left(\dfrac{-y}x\right)$. If the curve passes through the points $(1,2)$ and $(8,1)$, then $\left|y\left(\dfrac18\right)\right|$ is equal to

Let $y=y(x)$ be the solution of the differential equation
$$\frac{dy}{dx}=\frac{4y^3+2yx^2}{3xy^2+x^3},\quad y(1)=1.$$
If for some $n\in\mathbb N$, $y(2)\in[n-1,n)$, then $n$ is equal to ______.

If $\dfrac{dy}{dx}+2y\tan x=\sin x$, $0<x<\dfrac\pi2$ and $y\left(\dfrac\pi3\right)=0$, then the maximum value of $y(x)$ is :

Let a curve $y=y(x)$ pass through the point $(3,3)$ and the area of the region under this curve, above the $x$-axis and between the abscissae 3 and $x(>3)$ be $\left(\dfrac yx\right)^3$. If this curve also passes through the point $\left(\alpha,6\sqrt{10}\right)$ in the first quadrant, then $\alpha$ is equal to ______.

Let the solution curve $y=f(x)$ of the differential equation $\dfrac{dy}{dx}+\dfrac{xy}{x^2-1}=\dfrac{x^4+2x}{\sqrt{1-x^2}}$, $x\in(-1,1)$ pass through the origin. Then $\displaystyle\int_{-\frac{\sqrt3}2}^{\frac{\sqrt3}2}f(x)\,dx$ is equal to

Suppose $y=y(x)$ be the solution curve to the differential equation $\dfrac{dy}{dx}-y=2-e^{-x}$ such that $\displaystyle\lim_{x\to\infty}y(x)$ is finite. If $a$ and $b$ are respectively the $x$- and $y$-intercepts of the tangent to the curve at $x=0$, then the value of $a-4b$ is equal to ______.

Let $y=y_1(x)$ and $y=y_2(x)$ be two distinct solutions of the differential equation $\dfrac{dy}{dx}=x+y$, with $y_1(0)=0$ and $y_2(0)=1$ respectively. Then, the number of points of intersection of $y=y_1(x)$ and $y=y_2(x)$ is

Let $y=y(x)$ be the solution curve of the differential equation $\sin\left(2x^2\right)\log_e\left(\tan x^2\right)dy+\left(4xy-4\sqrt2x\sin\left(x^2-\dfrac\pi4\right)\right)dx=0$, $0<x<\sqrt{\dfrac\pi2}$, which passes through the point $\left(\sqrt{\dfrac\pi6},1\right)$. Then $\left|y\left(\sqrt{\dfrac\pi3}\right)\right|$ is equal to ______.

Consider a curve $y=y(x)$ in the first quadrant as shown in the figure. Let the area $A_1$ is twice the area $A_2$. Then the normal to the curve perpendicular to the line $2x-12y=15$ does NOT pass through the point.

Let $f$ be a differentiable function satisfying $f(x)=\dfrac2{\sqrt3}\displaystyle\int_0^{\sqrt3}f\left(\frac{\lambda^2x}3\right)d\lambda$, $x>0$ and $f(1)=\sqrt3$. If $y=f(x)$ passes through the point $(\alpha,6)$, then $\alpha$ is equal to ______.

Let the solution curve of the differential equation $x\,dy=\left(\sqrt{x^2+y^2}+y\right)dx$, $x>0$, intersect the line $x=1$ at $y=0$ and the line $x=2$ at $y=\alpha$. Then the value of $\alpha$ is :

If $y=y(x)$, $x\in(0,\pi/2)$ be the solution curve of the differential equation $\left(\sin^22x\right)\dfrac{dy}{dx}+\left(8\sin^22x+2\sin4x\right)y=2e^{-4x}(2\sin2x+\cos2x)$, with $y(\pi/4)=e^{-\pi}$, then $y(\pi/6)$ is equal to :