Differential Equations — JEE Main PYQs

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

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

Let $y = y(x)$ be the solution of the differential equation:

$$\frac{dy}{dx} + \left(\frac{6x^2 + \left(3x^2 + 2x^3 + 4\right)e^{-2x}}{\left(x^3 + 2\right)\left(2 + e^{-2x}\right)}\right)y = 2 + e^{-2x},$$

$x \in (-1, 2)$, satisfying $y(0) = \dfrac{3}{2}$. If $y(1) = \alpha(2 + e^{-2})$, $\alpha$ is equal to:

Let $y = y(x)$ be the solution of the differential equation

$$x \sin\left( \frac{y}{x} \right) dy = \left( y \sin\left( \frac{y}{x} \right) - x \right) dx, \quad y(1) = \frac{\pi}{2}$$

and let $\alpha = \cos\left( \frac{y\left( e^{12} \right)}{e^{12}} \right)$. Then the number of integral values of $p$, for which the equation $x^2 + y^2 - 2px + 2py + \alpha + 2 = 0$ represents a circle of radius $r \leq 6$, is ________.

Let $f : [1, \infty) \to \mathbf{R}$ be a differentiable function defined as $f(x) = \displaystyle\int_1^x f(t)\,dt + (1 - x)(\log_e x - 1) + e$. Then the value of $f(f(1))$ is :

Let $y = y(x)$ be the solution of the differential equation

$$(\tan x)^{1/2}\,dy = \left(\sec^3 x - (\tan x)^{3/2}\,y\right)dx, \; 0 < x < \frac{\pi}{2}, \; y\left(\frac{\pi}{4}\right) = \frac{6\sqrt{2}}{5}.$$

If $y\left(\dfrac{\pi}{3}\right) = \dfrac{4}{5}\,\alpha$, then $\alpha^4$ equals __________.

Let $y=y(x)$ be the solution of the differential equation

$$\left(x^2-x\sqrt{x^2-1}\right)dy+\left(y\left(x-\sqrt{x^2-1}\right)-x\right)dx=0,\ x \geq 1.$$

If $y(1)=1$, then the greatest integer less than $y\left(\sqrt{5}\right)$ is ________.

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

Let $y = y(x)$ be the solution of the differential equation

$$x\sqrt{1 - x^2}\, dy + \left(y\sqrt{1 - x^2} - x\cos^{-1}x\right)dx = 0,\; x \in (0, 1),\; \lim_{x \to 1^-} y(x) = 1.$$

Then $y\left(\frac{1}{2}\right)$ equals :

Let $f : \mathbf{R} \to \mathbf{R}$ be a thrice differentiable odd function satisfying $f'(x) \geq 0$, $f''(x) = f(x)$, $f(0) = 0$, $f'(0) = 3$. Then $9f(\log_\mathrm{e} 3)$ is equal to __________.

Let $f: [1, \infty) \to [2, \infty)$ be a differentiable function. If $10\displaystyle\int_1^x f(\mathrm{t})\,\mathrm{dt}=5xf(x)-x^5-9$ for all $x \geqslant 1$, then the value of $f(3)$ is :

Let $y=y(x)$ be the solution of the differential equation $\frac{\mathrm{d}y}{\mathrm{d}x}+2y\sec^2 x=2\sec^2 x+3\tan x \cdot \sec^2 x$ such that $y(0)=\frac{5}{4}$. Then $12\left(y\left(\frac{\pi}{4}\right)-\mathrm{e}^{-2}\right)$ is equal to _________.

Let $g$ be a differentiable function such that $\int_0^x g(t)\, dt = x - \int_0^x t\, g(t)\, dt$, $x \ge 0$ and let $y = y(x)$ satisfy the differential equation $\frac{dy}{dx} - y\tan x = 2(x+1)\sec x \ g(x)$, $x \in \left[0, \frac{\pi}{2}\right)$. If $y(0) = 0$, then $y\left(\frac{\pi}{3}\right)$ is equal to

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

Let $f : [0, \infty) \to \mathbb{R}$ be a differentiable function such that $f(x) = 1 - 2x + \int\limits_{0}^{x} e^{x-t} f(t)\, dt$ for all $x \in [0, \infty)$. Then the area of the region bounded by $y = f(x)$ and the coordinate axes is

If a curve $y=y(x)$ passes through the point $\left(1,\dfrac{\pi}{2}\right)$ and satisfies the differential equation $\left(7x^4\cot y-\mathrm{e}^x\operatorname{cosec} y\right)\dfrac{\mathrm{d}x}{\mathrm{d}y}=x^5$, $x\geq 1$, then at $x=2$, the value of $\cos y$ is :

Let $y = y(x)$ be the solution curve of the differential equation $x\left(x^2 + e^x\right) dy + \left(e^x (x - 2)\, y - x^3\right) dx = 0$, $x > 0$, passing through the point $(1, 0)$. Then $y(2)$ is equal to

Let $y = y(x)$ be the solution of the differential equation $(x^2 + 1)y' - 2xy = (x^4 + 2x^2 + 1)\cos x$, $y(0) = 1$. Then $\int\limits_{-3}^{3} y(x)\, \mathrm{d}x$ is :

Let $f(x) = x - 1$ and $g(x) = e^x$ for $x \in \mathbb{R}$. If $\frac{dy}{dx} = \left(e^{-2\sqrt{x}}\, g\left(f\left(f(x)\right)\right) - \frac{y}{\sqrt{x}}\right)$, $y(0) = 0$, then $y(1)$ is

Let $x=x(y)$ be the solution of the differential equation $y^2 dx+\left(x-\frac{1}{y}\right)dy=0$. If $x(1)=1$, then $x\left(\frac{1}{2}\right)$ is :

If $x = f(y)$ is the solution of the differential equation

$$\left(1 + y^2\right) + \left(x - 2e^{\tan^{-1} y}\right) \dfrac{dy}{dx} = 0, \; y \in \left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$$

with $f(0) = 1$, then $f\left(\dfrac{1}{\sqrt{3}}\right)$ is equal to :

Let $y = f(x)$ be the solution of the differential equation $\dfrac{dy}{dx} + \dfrac{xy}{x^2 - 1} = \dfrac{x^6 + 4x}{\sqrt{1 - x^2}}$, $-1 < x < 1$ such that $f(0) = 0$. If $6\displaystyle\int_{-1/2}^{1/2} f(x)\,dx = 2\pi - \alpha$ then $\alpha^2$ is equal to ________.

Let a curve $y = f(x)$ pass through the points $(0, 5)$ and $(\log_e 2, k)$. If the curve satisfies the differential equation $2(3 + y)e^{2x}dx - (7 + e^{2x})dy = 0$, then $k$ is equal to

Let $x = x(y)$ be the solution of the differential equation $y = \left(x - y \dfrac{\mathrm{d}x}{\mathrm{d}y}\right) \sin\left(\dfrac{x}{y}\right)$, $y > 0$ and $x(1) = \dfrac{\pi}{2}$. Then $\cos(x(2))$ is equal to :

Let $y = y(x)$ be the solution of the differential equation $\left(xy - 5x^2\sqrt{1+x^2}\right)dx + (1+x^2)dy = 0$, $y(0) = 0$. Then $y(\sqrt{3})$ is equal to

Let f be a differentiable function such that $2(x+2)^2 f(x) - 3(x+2)^2 = 10\int_0^x (t+2)f(t)\,dt$, $x \geq 0$. Then $f(2)$ is equal to _____.

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

Let $y = y(x)$ be the solution of the differential equation $2\cos x \dfrac{\mathrm{d}y}{\mathrm{d}x} = \sin 2x - 4y\sin x$, $x \in \left(0, \dfrac{\pi}{2}\right)$. If $y\left(\dfrac{\pi}{3}\right) = 0$, then $y'\left(\dfrac{\pi}{4}\right) + y\left(\dfrac{\pi}{4}\right)$ is equal to __________.

Let for some function $y = f(x)$, $\int_0^x t\, f(t)\, dt = x^2 f(x)$, $x > 0$ and $f(2) = 3$. Then $f(6)$ is equal to

If $y = y(x)$ is the solution of the differential equation, $\sqrt{4 - x^2}\, \frac{\mathrm{d}y}{\mathrm{d}x} = \left(\left(\sin^{-1}\left(\frac{x}{2}\right)\right)^2 - y\right) \sin^{-1}\left(\frac{x}{2}\right)$, $-2 \le x \le 2$, $y(2) = \frac{\pi^2 - 8}{4}$, then $y^2(0)$ is equal to __________.

If the solution $y=y(x)$ of the differential equation $(x^{4}+2x^{3}+3x^{2}+2x+2)dy-(2x^{2}+2x+3)dx=0$ satisfies $y(-1)=-\frac{\pi}{4}$, then $y(0)$ 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 ________.

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

Let $y = y(x)$ be the solution of the differential equation $(x + y + 2)^2 dx = dy$, $y(0) = -2$. Let the maximum and minimum values of the function $y = y(x)$ in $\left[0, \dfrac{\pi}{3}\right]$ be $\alpha$ and $\beta$, respectively. If $(3\alpha + \pi)^2 + \beta^2 = \gamma + \delta\sqrt{3}, \gamma, \delta \in \mathbb{Z}$, then $\gamma + \delta$ equals ________

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

Let $f$ be a differentiable function in the interval $(0, \infty)$ such that $f(1) = 1$ and $\displaystyle\lim_{t \to x} \frac{t^2 f(x) - x^2 f(t)}{t - x} = 1$ for each $x > 0$. Then $2f(2) + 3f(3)$ is equal to ________.

The differential equation of the family of circles passing through the origin and having centre at the line $y = x$ is :

Let $y = y(x)$ be the solution of the differential equation

$$\dfrac{\mathrm{d}y}{\mathrm{d}x} + \dfrac{2x}{(1 + x^2)^2}\,y = x\,\mathrm{e}^{\frac{1}{(1 + x^2)}};\ y(0) = 0.$$

Then the area enclosed by the curve $f(x) = y(x)\,\mathrm{e}^{-\frac{1}{(1 + x^2)}}$ and the line $y - x = 4$ is ________.

Let $y = y(x)$ be the solution of the differential equation $\left(1 + x^2\right) \dfrac{dy}{dx} + y = e^{\tan^{-1} x}$, $y(1) = 0$. Then $y(0)$ is

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

Suppose the solution of the differential equation $\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{(2+\alpha)x-\beta y+2}{\beta x-2\alpha y-(\beta\gamma-4\alpha)}$ represents a circle passing through origin. Then the radius of this circle is :

If the solution $y(x)$ of the given differential equation $(\mathrm{e}^y+1)\cos x\,\mathrm{d}x+\mathrm{e}^y \sin x\,\mathrm{d}y=0$ passes through the point $\left(\frac{\pi}{2}, 0\right)$, then the value of $\mathrm{e}^{y\left(\frac{\pi}{6}\right)}$ 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

Let $y=y(x)$ be the solution of the differential equation $(1+y^2)e^{\tan x}\,dx+\cos^2 x\,(1+e^{2\tan x})\,dy=0$, $y(0)=1$. Then $y\left(\frac{\pi}{4}\right)$ is equal to

Let $y = y(x)$ be the solution curve of the differential equation $\sec y \dfrac{\mathrm{d}y}{\mathrm{d}x} + 2x\sin y = x^3\cos y$, $y(1) = 0$. Then $y(\sqrt{3})$ is equal to :

Let $\alpha|x| = |y|\mathrm{e}^{xy - \beta}$, $\alpha, \beta \in \mathbb{N}$ be the solution of the differential equation $x\mathrm{d}y - y\mathrm{d}x + xy(x\mathrm{d}y + y\mathrm{d}x) = 0$, $y(1) = 2$. Then $\alpha + \beta$ is equal to ________

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

The solution curve, of the differential equation $2y\,\frac{\mathrm{d}y}{\mathrm{d}x}+3=5\,\frac{\mathrm{d}y}{\mathrm{d}x}$, passing through the point $(0, 1)$ is a conic, whose vertex lies on the line :

Let $\int_{0}^{x}\sqrt{1-\left(y'(t)\right)^2}\,dt=\int_{0}^{x} y(t)\,dt$, $0 \le x \le 3$, $y \ge 0$, $y(0)=0$. Then at $x=2$, $y''+y+1$ is equal to

For a differentiable function $f: \mathbb{R} \to \mathbb{R}$, suppose $f'(x)=3f(x)+\alpha$, where $\alpha \in \mathbb{R}$, $f(0)=1$ and $\lim\limits_{x \to -\infty} f(x)=7$. Then $9f\left(-\log_e 3\right)$ is equal to ________.

Let $y = y(x)$ be the solution of the differential equation $\dfrac{dy}{dx} = 2x(x+y)^3 - x(x+y) - 1$, $y(0) = 1$.

Then, $\left(\dfrac{1}{\sqrt{2}} + y\left(\dfrac{1}{\sqrt{2}}\right)\right)^2$ equals :

If $x = x(t)$ is the solution of the differential equation $(t + 1)dx = \left(2x + (t + 1)^4\right)dt$, $x(0) = 2$, then, $x(1)$ equals ________.

Let $\alpha$ be a non-zero real number. Suppose $f: \mathbf{R} \to \mathbf{R}$ is a differentiable function such that $f(0)=2$ and $\lim_{x \to -\infty} f(x)=1$. If $f'(x)=\alpha f(x)+3$, for all $x \in \mathbf{R}$, then $f(-\log_e 2)$ is equal to ________.

If $\frac{\mathrm{d}x}{\mathrm{d}y}=\frac{1+x-y^2}{y}$, $x(1)=1$, then $5x(2)$ is equal to __________.

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, $a, b \in \mathbf{R}$. Given that $x(0) = 2$; $y(0) = 1$ and $3y(1) = 2x(1)$, the value of $t$, for which $x(t) = y(t)$, is :

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 ________.

A function $y=f(x)$ satisfies $f(x)\sin 2x + \sin x - \left(1+\cos^2 x\right)f'(x) = 0$ with condition $f(0)=0$. Then, $f\left(\dfrac{\pi}{2}\right)$ is equal to

If the solution curve $y=y(x)$ of the differential equation $(1+y^2)(1+\log_e x)\,dx + x\,dy = 0$, $x > 0$ passes through the point $(1,1)$ and $y(e) = \dfrac{\alpha - \tan\left(\dfrac{3}{2}\right)}{\beta + \tan\left(\dfrac{3}{2}\right)}$, then $\alpha + 2\beta$ is __________.