Differential Equations Questions - JEE

The orthogonal trajectories of $y = cx^2$ are:

\$y''-4y'+4y=0\$, \$y(0)=1\$, \$y'(0)=2\$. Find \$y(1)\$.

If $y=y(x)$ is the solution of the differential equation $\dfrac{dy}{dx}+2y=\sin(2x)$, $y(0)=\dfrac{3}{4}$, then $y\left(\dfrac{\pi}{8}\right)$ is equal to:

If \$\\dfrac{dy}{dx}=\\dfrac{ax+3}{2y+f}\$ represents a circle, find \$a+f\$.

The perpendicular from the origin to the tangent at any point on a curve is equal to the abscissa of the point of contact. Also curve passes through the point $(1, 1)$. Then the length of intercept of the curve on the x-axis is ______.

Let $y = f(x)$ be a curve $C_1$ passing through $(2,2)$ and $\left(8, \frac{1}{2}\right)$ and satisfying a differential equation $y\left(\frac{d^2y}{dx^2}\right) = 2\left(\frac{dy}{dx}\right)^2$. Curve $C_2$ is the director circle of the circle $x^2 + y^2 = 2$. If the shortest distance between the curves $C_1$ and $C_2$ is $\left(\sqrt{p-q}\right)$ where $p,q \in \mathbb{N}$, then find the value of $\left(p^2 - q\right)$.

Solution of the differential equation $x\cos\left(\frac{x}{y}\right)(ydx + xdy) = y\sin\left(\frac{x}{y}\right)(xdy - ydx)$ is :

Let $y(x)$ be a function satisfying $d^2y/dx^2 - dy/dx + e^{2x} = 0.5, y(0) = 2$ and $y'(0) = 1$. If maximum value of $y(x)$ is $y(a)$. Then integral part of $(2a)$ is ____.

We have $y - (c_1 + c_2)\sin(x + c_3) - c_4 e^{x+c_5}$. The order of the differential equation whose general solution is given by this expression is:

$y = f(x)$ is a particular solution of differential equation $\left(\frac{dy}{dx}\right)^2 + y\frac{d^2y}{dx^2} = e^x$ such that $f(0) = 1 = f'(0)$, then the value of $f^2(\ln 2)$ is ____.

If $\frac{dy}{dx} = \frac{x^2-y}{x+y}$ such that $y = f(x)$ is a solution of differential equation & $f(0) = 0$, then:

If $y=y(x)$ is the solution curve of the differential equation $\dfrac{dy}{dx}+y\tan x=x\sec x$, $0\leq x\leq\dfrac{\pi}{3}$, $y(0)=1$, then $y\!\left(\dfrac{\pi}{6}\right)$ is equal to:

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

Identify the statement(s) which is/are true?

Let 759. Let $A = \int_0^2 y\,dx$ and the differential equation $\dfrac{dy}{y+A} = dx$ with $y(0)=1$. Find $y(x)$.

If differential equation of first degree of a curve is given by $x^2\left(\frac{dy}{dx}\right)^2 - x(2y - 1)\frac{dy}{dx} + (y^2 - y - 2) = 0$, then $(y - 2015 . x)$ is a positive prime number '$P$' then the value of $P$ is ____.

Let $\frac{d}{dx}f(x) = \frac{e^{\sin x}}{x}$, $x > 0$. If $\int_1^{2e^{\sin x^2}} \frac{2e^{\sin x^2}}{x}dx = F(k) - F(1)$, then find one of the possible value of $k/4$.

The differential equation of all conics whose centre lies at origin, is given by

Find the equation of the curve passing through (1, 2) whose differential equation is y(x + y³)dx = x(y³ - x)dy.

Tangent is drawn at the point $(x_i, y_i)$ on the curve $y = f(x)$, which intersects the x-axis at $(x_{i+1}, 0)$. Now, again a tangent is drawn at $(x_{i+1}, y_{i+1})$ on the curve which intersects the x-axis at $(x_{i+2}, 0)$ and the process is repeated $n$ times, i.e., $i = 1, 2, 3, ........., n$. If $x_1, x_2, x_3, ........., x_n$ form an arithmetic progression with common difference equal to $\log_2 e$ and curve passes through $(0, 2)$. Now if curve passes through the point $(-2, k)$, then the value of $k$ is ______.

The differential equation corresponding to the family of curves $x^2 + y^2 - 2ay = 0$, where $a$ is an arbitrary constant, is:

Let $y = y(x)$ be the solution of the differential equation $\sin x\frac{dy}{dx} + y\cos x = 4x,\ x \in (0,\pi)$. If $y\!\left(\frac{\pi}{2}\right) = 0$, then $y\!\left(\frac{\pi}{6}\right)$ is equal to

The solution of the differential equation $y\,dx + (x + x^2 y)\,dy = 0$ is:

99. The solution of the differential equation $e^{-x}(y+1)\, dy + (\cos^2 x - \sin 2x)\, y\, dx = 0$ subjected to condition that $y = 1$ when $x = 0$, is:

The solution of the differential equation $(1 + y^2) + (x - e^{\tan^{-1}y})\frac{dy}{dx} = 0$ is:

The differential equation $\frac{dy}{dx} = \frac{(x + y + 2)^2 - (y - 1)^2}{(x + 3)^2}$ determines a curve $y = f(x)$ & $f(0) = 1$ then $f(3)$ is equal to ____.

Let $y=y(x)$ be the solution of the differential equation $(3y^2-5x^2)y\,dx+2x(x^2-y^2)\,dy=0$ such that $y(1)=1$. Then $\left|(y(2))^3-12y(2)\right|$ is equal to:

A curve passes through the point (0,1) and the gradient at (x, y) on it is $y(xy - 1)$. The equation of the curve is

Given $(y^2 - x^3)\,dx - xy\,dy = 0$. Find the solution of the differential equation.

97. The general solution of the differential equation $(1 + \tan y)(dx - dy) + 2x\, dy = 0$ is:[Note: Where $C$ is constant of integration.]

Solve $y = x\frac{dy}{dx} + \left(\frac{dy}{dx}\right)^2$.

If $y = y(x)$ is the solution of the differential equation $x\dfrac{dy}{dx} + 2y = x^2$ satisfying $y(1) = 1$, then $y\!\left(\dfrac{1}{2}\right)$ is equal to

If the dependent variable $y$ is changed to $z$ by the substitution $y = \tan z$ and the differential equation $\frac{d^2y}{dx^2} = 1 + \frac{2(1+y^2)}{1+y^2}\left(\frac{dy}{dx}\right)^2$ is changed to $\frac{d^2z}{dx^2} = \cos^2 z + k\left(\frac{dz}{dx}\right)^2$, then the value of $k$ equals ______.

If the solution of the differential equation $\frac{dy}{dx} - y = 1 - e^{-x}$ and $y(0) = y_0$ has a finite value, when $x \to \infty$, then the value of $|2/y_0|$ is ______.

If $y = y(x)$ is the solution of the differential equation $\frac{dy}{dx} = (\tan x - y)\sec^2 x,\ x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, such that $y(0) = 0$, then $y\!\left(-\frac{\pi}{4}\right)$ is equal to:

If $\dfrac{dy}{dx} = y + 3 > 0$ and $y(0) = 2$, then $y(\ln 2)$ is equal to

If $\phi(x)$ is a differential real-valued function satisfying $\phi'(x) + 2\phi(x) \leq 1$, then the maximum value of $2\phi(x)$, equal to ____.

The differential equation of all circles passing through the origin and having their centres on the x-axis is

If the differential equation representing the family of all circles touching x-axis at the origin is $(x^2 - y^2)\dfrac{dy}{dx} = g(x)\, y$, then $g(x)$ equals

The solution of the differential equation $x^4\frac{dy}{dx} + x^3y + \csc(xy) = 0$ is equal to

A tank initially contains 50 gallons of fresh water. Brine contains 2 pounds per gallon of salt, flows into the tank at the rate of 2 gallons per minutes and the mixture kept uniform by stirring, runs out at the same rate. If it will take for the quantity of slat in the tank to increase from 40 to 80 pounds (in seconds) is $206x$, then find $x$. (given $\ln 3 = 1.0986$)

The differential equation representing the family of ellipses having foci either on the x-axis or on the y-axis, centre at the origin and passing through the point (0, 3) is

98. The solution of the differential equation $y^2\, dx + (x^2 - xy + y^2)\, dy = 0$, is:[Note: Where $C$ is constant of integration.]

Let the solution curve $y=y(x)$ of the differential equation $\dfrac{dy}{dx}-\dfrac{3x^5\tan^{-1}(x^3)}{(1+x^6)^{3/2}}y=2x\exp\!\left(\dfrac{x^3-\tan^{-1}x^3}{\sqrt{(1+x)^6}}\right)$ pass through the origin. Then $y(1)$ is equal to:

Question 21: If $\frac{f'(x)}{f(x)} = 1$ and $f(1) = 2$, then find $\left[\frac{f(3)}{2}\right]$, where $[\cdot]$ denotes the greatest integer function.

If the solution $y(x)$ of the given differential equation $(e^y+1)\cos x\,dx+e^y\sin x\,dy=0$ passes through the point $\left(\dfrac{\pi}{2},0\right)$, then the value of $e^{y(\pi/6)}$ is equal to ________.

The equation of the curve passing through $(3, 4)$ and satisfying the differential equation $y\frac{dy}{dx} + (x^2 - y)\frac{dy}{dx} - x^2 = 0$ can be

Let $y(x)$ be the solution of the differential equation $(x\log x)\frac{dy}{dx} + y = 2x\log x,\ (x \geq 1)$. Then $y(e)$ is equal to

Use the methods of solving first order differential equation to find the general solution of $\frac{d^2y}{dx^2} = 3\left(\frac{dy}{dx}\right)^{3/2}$.

If $m$ and $n$ are order and degree of the equation $2\left(\frac{d^2y}{dx^2}\right)^4 + 3\frac{d^2y}{dx^2}\left(\frac{d^3y}{dx^3}\right)^5 = x^2 - 1$, then

Differential Equations Questions (544)