JEE Differential Equations PYQs With Video Solutions PDF

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 $$y = y(x)$$ 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 $$|(y(2))^3 - 12y(2)|$$ is equal to:

Let the tangent at any point P on a curve passing through the points (1, 1) and ($$\frac{1}{10}$$, 100), intersect positive x-axis and y-axis at the points A and B respectively. If PA : PB = 1 : k and $$y = y(x)$$ is the solution of the differential equation $$e^{\frac{dy}{dx}} = kx + \frac{k}{2}$$, $$y(0) = k$$, then $$4y(1) - 5\log_e 3$$ is equal to _______.

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 $$\left(0, \dfrac{1}{2} + \dfrac{\pi}{2\sqrt{2}}\right)$$ and $$\left(\alpha, \dfrac{1}{2}e^{2\alpha}\right)$$ then $$e^{\alpha}$$ is equal to

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 $$\left|y\left(\dfrac{1}{8}\right)\right|$$ is equal to

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 $$\left(\dfrac{y}{x}\right)^3$$. If this curve also passes through the point $$(\alpha, 6\sqrt{10})$$ in the first quadrant, then $$\alpha$$ is equal to ______.

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

The population $$P = P(t)$$ at time $$t$$ of a certain species follows the differential equation $$\frac{dP}{dt} = 0.5P - 450$$. If $$P(0) = 850$$, then the time at which population becomes zero is:

If $$y = y(x)$$ is the solution of the differential equation, $$\frac{dy}{dx} + 2y\tan x = \sin x$$, $$y\left(\frac{\pi}{3}\right) = 0$$, then the maximum value of the function $$y(x)$$ over $$R$$ is equal to:

Let $$f: \mathbb{R} \to \mathbb{R}$$ be such that $$f(xy) = f(x)f(y)$$, for all $$x, y \in \mathbb{R}$$ and $$f(0) \ne 0$$. Let $$g: [1, \infty) \to \mathbb{R}$$ be a differentiable function such that $$x^2 g(x) = \int_1^x (t^2 f(t) - tg(t))\,dt.$$ Then $$g(2)$$ is equal to :

Let $$y = y(x)$$ be the solution of the differential equation $$x \tan\left(\frac{y}{x}\right) dy = \left(y \tan\left(\frac{y}{x}\right) - x\right) dx$$, $$-1 \le x \le 1$$, $$y\left(\frac{1}{2}\right) = \frac{\pi}{6}$$. Then the area of the region bounded by the curves $$x = 0$$, $$x = \frac{1}{\sqrt{2}}$$ and $$y = y(x)$$ in the upper half plane is:

If $$x\phi(x) = \int_5^x (3t^2 - 2\phi'(t)) dt$$, $$x > -2$$, $$\phi(0) = 4$$, then $$\phi(2)$$ is _________.

Which of the following is true for $$y(x)$$ that satisfies the differential equation $$\frac{dy}{dx} = xy - 1 + x - y$$; $$y(0) = 0$$:

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

Let $$y = y(x)$$ be a solution curve of the differential equation $$(y+1)\tan^2 x \, dx + \tan x \, dy + y \, dx = 0$$, $$x \in \left(0, \frac{\pi}{2}\right)$$. If $$\lim_{x \to 0^+} xy(x) = 1$$, then the value of $$y\left(\frac{\pi}{4}\right)$$ is:

The rate of growth of bacteria in a culture is proportional to the number of bacteria present and the bacteria count is 1000 at initial time $$t = 0$$. The number of bacteria is increased by 20% in 2 hours. If the population of bacteria is 2000 after $$\frac{k}{\log_e(\frac{6}{5})}$$ hours, then $$\left(\frac{k}{\log_e 2}\right)^2$$ is equal to:

Let $$\alpha x = \exp(x^\beta y^\gamma)$$ be the solution of the differential equation $$2x^2 y dy - (1 - xy^2)dx = 0$$, $$x > 0, y(2) = \sqrt{\log_e 2}$$. Then $$\alpha + \beta - \gamma$$ equals :

The difference between degree and order of a differential equation that represents the family of curves given by $$y^2 = a\left(x + \frac{\sqrt{a}}{2}\right), a > 0$$ is ______.

If $$\frac{dx}{dy} = \frac{1 + x - y^2}{y}$$, $$x(1) = 1$$, then $$5x(2)$$ is equal to:

Let  $$f:(0,\infty)\to R$$ be a function which is differentiable at all points of its domain and satisfies the condition $$x^2 f'(x) = 2x f(x) + 3,$$  with  $$f(1)=4.$$ Then  $$2f(2)$$  is equal to:

Let $$y = y(x)$$ be the solution of the differential equation $$\cos x\frac{dy}{dx} + 2y\sin x = \sin 2x$$, $$x \in \left(0, \frac{\pi}{2}\right)$$. If $$y(\pi/3) = 0$$, then $$y(\pi/4)$$ is equal to:

If $$y = y(x)$$ is the solution of the differential equation $$\frac{dy}{dx} + \frac{4x}{x^2-1}y = \frac{x+2}{(x^2-1)^{5/2}}$$, $$x \gt 1$$ such that $$y(2) = \frac{2}{9}\log_e 2 + \sqrt{3}$$ and $$y\sqrt{2} = \alpha\log_e(\sqrt{\alpha} + \beta) + \beta - \sqrt{\gamma}$$, $$\alpha, \beta, \gamma \in \mathbb{N}$$, then $$\alpha\beta\gamma$$ is equal to _____.

Let a differentiable function $$f$$ satisfy $$f(x) + \int_3^x \dfrac{f(t)}{t} dt = \sqrt{x+1}$$, $$x \ge 3$$. Then $$12f(8)$$ is equal to:

Let $$y = y(x)$$ be the solution of the differential equation $$(1 - x^2)dy = \left[xy + (x^3 + 2)\sqrt{3(1 - x^2)}\right]dx$$, $$-1 < x < 1$$, $$y(0) = 0$$. If $$y\left(\frac{1}{2}\right) = \frac{m}{n}$$, $$m$$ and $$n$$ are coprime numbers, then $$m + n$$ is equal to __________.

Let $$f(x)$$ be a positive function such that the area bounded by $$y = f(x), y = 0$$ from $$x = 0$$ to $$x = a > 0$$ is $$e^{-a} + 4a^2 + a - 1$$. Then the differential equation, whose general solution is $$y = c_1 f(x) + c_2$$, where $$c_1$$ and $$c_2$$ are arbitrary constants, is

If the solution of the differential equation $$(2x + 3y - 2)dx + (4x + 6y - 7)dy = 0$$, $$y(0) = 3$$, is $$\alpha x + \beta y + 3\log_e|2x + 3y - \gamma| = 6$$, then $$\alpha + 2\beta + 3\gamma$$ is equal to _______.

Let $$y = y(x)$$ be the solution of the differential equation $$\frac{dy}{dx} + \frac{2x}{(1+x^2)^2} y = xe^{\frac{1}{(1+x^2)}}$$; $$y(0) = 0$$. Then the area enclosed by the curve $$f(x) = y(x)e^{-\frac{1}{(1+x^2)}}$$ and the line $$y - x = 4$$ is __________

Let $$y = y(x)$$ be the solution of the differential equation $$(x^2 + 4)^2 dy + (2x^3 y + 8xy - 2)dx = 0$$. If $$y(0) = 0$$, then $$y(2)$$ is equal to

Let the solution $$y = y(x)$$ of the differential equation $$\frac{dy}{dx} - y = 1 + 4\sin x$$ satisfy $$y(\pi) = 1$$. Then $$y\left(\frac{\pi}{2}\right) + 10$$ is equal to ______.

If for the solution curve y = f(x) of the differential equation $$\frac{dy}{dx}+(\tan x)y = \frac{2+\sec x}{(1+2\sec x)^{2}}$$, $$x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right), \quad f\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{10}$$, then $$f\left(\frac{\pi}{4}\right)$$ is equal to :