Differential Equations Questions - JEE

Let $y=y(x)$ be a solution of $\dfrac{dy}{dx} + \dfrac{y\cos x}{\sin x+\cos x} = \dfrac{2\tan^2 x+\tan x+2}{\sqrt{\sin x+\cos x}}$. If $y\!\left(\dfrac{\pi}{4}\right)=2^{3/4}$, then $y^2\!\left(\dfrac{7\pi}{12}\right)$ is

Given $x\dfrac{dy}{dx} + 2y = x^2$ where $x \neq 0$ and $y(1) = 1$. Find $y$.

If $\dfrac{dy}{dx} + y = xe^{-x}$ and $C = 0$, then $f(1)$ equals:

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

Consider the differential equation, $y^2 dx + \left(x - \dfrac{1}{y}\right)dy = 0$. If value of $y$ is 1 when $x = 1$, then the value of $x$ for which $y = 2$, is:

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

If $\frac{dy}{dx} + y\tan x = \sin 2x$ and $y(0) = 1$, then $y(\pi)$ is equal to

If $y = f(x)$ satisfies the differential equation $\sin x\,\dfrac{dy}{dx} + 2y\cos x = 8$ with $f(\pi/2) = 8$, then the minimum value of $f(x)$ is:

If y₁, y₂ are two solutions of the differential equation dy/dx + P(x)·y = Q(x), then prove that y = y₁ + c(y₁ - y₂) is the general solution of the equation where c is any constant. For what relation between the constants α, β will the linear combination αy₁ + βy₂ also be a solution?

Given that the differential equation is $\dfrac{d^2y}{dx^2} = e^{-2x}$. The general solution is:

$f : R \to R$ be a twice differentiable function satisfying $f''(x) - 5f'(x) + 6f(x) \geq 0$ $\forall\, x \geq 0$ if $f(0) = 1$, $f'(0) = 0$. If $f(x)$ satisfies $f(x) \geq ah(bx) - bh(ax)$, $\forall\, x \geq 0$, then find $(a+b)h(0)/2$.

The differential equation of the family of non-vertical lines $y = mx + c$, where $m$ and $c$ are two parameters, is:

Consider the differential equation\[ 2y\frac{dy}{dx} + y^2 \sec x = \tan x \]Which of the following is true about the method of solving this equation?

The solution of the differential equation $y'' = \dfrac{1}{(xy'-y)}(xy''+y'-y')$ is:

Let the population of rabbits surviving at a time $t$ be governed by the differential equation $\dfrac{dp(t)}{dt} = \dfrac{1}{2}p(t) - 200$. If $p(0) = 100$, then $p(t)$ equals

If $y = (x + \sqrt{1+x^2})^n$, then $(1+x^2)\dfrac{d^2y}{dx^2} + x\dfrac{dy}{dx}$ is

The equation of normal at $(x, y)$ to a curve is \[Y - y = \frac{-dx}{dy}(X - x)\] Given $\dfrac{1}{OA} + \dfrac{1}{OB} = 1$, where $OA = x + y\dfrac{dy}{dx}$ and $OB = \dfrac{x + y\dfrac{dy}{dx}}{\dfrac{dy}{dx}}$. Find the answer. (Answer: 0.80)

The solution of differential equation $\dfrac{dy}{dx} = \dfrac{x^2 + y^2 + 1}{2xy}$ satisfying $y(1) = 0$ is given by:

Let $y = y(x)$ be the solution of the differential equation, $x\frac{dy}{dx} + y = x\log_e x,\ (x > 1)$. If $2y(2) = \log_e 4 - 1$, then $y(e)$ is equal to:

The slope of tangent at any point $(x,y)$ on a curve $y=y(x)$ is $\dfrac{x^2+y^2}{2xy}$, $x>0$. If $y(2)=0$, then a value of $y(8)$ is

Let $y = f(x)$ be a differentiable function satisfying $f(x) + f'(x) = xe^{-x}$ for all values of real $x$. If $f(0) = 0$, then the value of $f(1)$ equals:

The rate of cooling of a substance in moving air is proportional to the difference of temperatures of the substance and the air. A substance cools from 36°C to 34°C in 15 minutes. Find when the substance will have the temperature 32°C, it being known that the constant temperature of air is 30°C.

On applying Newton–Leibniz rule to the equation\[ x[y(x) - 0] + \int_1^x y(t)\,dt = \int_1^x ty(t)\,dt + (x+1)(xy(x)-0) \]and simplifying, the solution $ y(x) $ is found to be:

The general equation of circles touching the y-axis at the origin is $(x+g)^2 + y^2 = g^2$. The differential equation of such circles is:

The solution of $3x(1 - x^2)y^2\frac{dy}{dx} + (2x^2 - 1)y^3 = ax^3$ is

Let $f$ be a twice differentiable non-negative function such that $\left(f(x)\right)^2=25+\displaystyle\int_0^x\!\left(\left(f(t)\right)^2+\left(f'(t)\right)^2\right)dt$. Then the mean of $f\!\left(\log_e 1\right),\,f\!\left(\log_e 2\right),\,\ldots,\,f\!\left(\log_e 625\right)$ is equal to _____.

Population doubles in 5 years (\$\\frac{dP}{dt}=kP\$). How many years to become 8 times?

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

If the solution curve $y=f(x)$ of the differential equation $\left(x^2-4\right)y'-2xy+2x\left(4-x^2\right)^2=0$, $x>2$, passes through the point $(3,15)$, then the local maximum value of $f$ is _____.

\$y'-y\\tan x=2x\\sec x\$, \$y(0)=0\$. Find \$y(\\pi/4)\$.

The solution of x\frac{dy}{dx} + y = y^2 \log x is

Degree of \$\\left[1+\\left(\\dfrac{dy}{dx}\\right)^2\\right]^{3/2} = \\dfrac{d^2y}{dx^2}\$.

The curve for which the intercept cut off by any tangent on Y-axis is proportional to the square of the ordinate of the point of tangency is

\$(2+\\sin x)\\dfrac{dy}{dx}+(y+1)\\cos x=0\$, \$y(0)=1\$. Find \$y(\\pi/2)\$.

Which curve represents the solution of the IVP $\dfrac{dy}{dx} = 100 - y$, where $y(0) = 50$?

If $\frac{dy}{dx} + \frac{3}{\cos^2 x}y = \frac{1}{\cos^2 x},\ x \in \left(\frac{-\pi}{3}, \frac{\pi}{3}\right)$, and $y\!\left(\frac{\pi}{4}\right) = \frac{4}{3}$, then $y\!\left(-\frac{\pi}{4}\right)$ equals:

The solution of the differential equation (x - y) dy - (x + y) dx = 0 is

Solution of the differential equation (x2 − ay)dx − (ax − y2)dy = 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 (JEE Main 2019)

The solution of $\frac{dy}{dx} + 2y\cot x = y^2$ is

The function $f(x)$ is defined for $x \geq 0$ and has its inverse $g(x)$ which is differentiable. If $f(x)$ satisfies $\int_0^x f(t)\,dt = x^2$ and $g(0) = 0$ then

Solve xdy - ydx - (y + xy²)(1 + log x)dx = 0.

Given $(x^2+1)^2 \dfrac{dy}{dx} + 2x(x^2+1)y = 1$ with $y(0) = 0$. If $\sqrt{a}\, y(1) = \dfrac{\pi}{32}$, find the value of $a$.

A function $f: \mathbb{R} \to \mathbb{R}$ satisfies $\sin x \cos y (f(2x + 2y) − f(2x − 2y)) = \cos x \sin y (f(2x + 2y) + f(2x − 2y))$. If $f(0) = \frac{1}{2}$, find the differential equation that $f$ satisfies.

Curve $y=f(x)$ through origin satisfies $\dfrac{7dy}{dx}+\dfrac{28x^3y}{1+x^4}=\dfrac{40x^4}{1+x^4}$. If area enclosed by $y=f^{-1}(x)$, x-axis, and $x=4/7$ in 1st quadrant is $A$, then $42A$ is

The solution of the differential equation $\dfrac{dy}{dx} = \dfrac{x+y}{x}$ satisfying the condition $y(1) = 1$ is

The general solution of $\dfrac{dy}{dx} = \dfrac{1+x+y+xy}{1-x}$ is a family of curves which looks most like:

If a curve is such that line joining origin to any point $P(x, y)$ on the curve and the line parallel to $y$-axis through $P$ are equally inclined to tangent to curve at $P$, then the differential equation of the curve is:

Let $x=x(t)$ and $y=y(t)$ be solutions of the differential equations $\dfrac{dx}{dt}+ax=0$ and $\dfrac{dy}{dt}+by=0$ respectively, $a,b\in\mathbb{R}$. Given that $x(0)=2$; $y(0)=1$ and $3y(1)=2x(1)$, the value of $t$, for which $x(t)=y(t)$, is:

The solution curve of the differential equation $y\,\dfrac{dx}{dy}=x\left(\log_e x-\log_e y+1\right)$, $x>0$, $y>0$ passing through the point $(e,1)$ is

Differential Equations Questions (544)