Differential Equations-JEE-Mains-PYQ’s

Mathematics Questions

Let $y = y(x)$ be the solution of the differential equation $x \frac{dy}{dx} - y = x^2 \cot x, x \in (0, \pi)$ . If $y\left(\frac{\pi}{2}\right) = \frac{\pi}{2}$ , then $6y\left(\frac{\pi}{6}\right) - 8y\left(\frac{\pi}{4}\right)$ is equal to :

(A) $-3\pi$

(B) $3\pi$

(D) $\pi$

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Let $y = y(x)$ be the solution of the differential equation $x \frac{dy}{dx} - \sin 2y = x^3 (2 - x^3) \cos^2 y, x \neq 0$ . If $y(2) = 0$ , then $\tan(y(1))$ is equal to :

(A) $-\frac{7}{4}$

(B) $-\frac{3}{4}$

(D) $\frac{7}{4}$

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

(A) $92$

(B) $72$

(D) $81$

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If $y = y(x)$ satisfies the differential equation $16(\sqrt{x} \cdot \sqrt{9 + \sqrt{x}})(4 + \sqrt{9 + \sqrt{x}}) \cos y \, dy = (1 + 2 \sin y) dx, x > 0$ and $y(256) = \frac{\pi}{2}, y(49) = \alpha$ , then $2 \sin \alpha$ is equal to :

(A) $2\sqrt{2} - 1$

(B) $\sqrt{2} - 1$

(D) $3(\sqrt{2} - 1)$

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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 :

(A) $6$

(B) $4$

(D) $2$

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Let $y = y(x)$ be the solution of the differential equation $\sec x \frac{dy}{dx} - 2y = 2 + 3 \sin x, x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right), y(0) = -\frac{7}{4}$ . Then $y\left(\frac{\pi}{6}\right)$ is equal to :

(A) $-\frac{5}{2}$

(B) $-3\sqrt{2} - 7$

(D) $-3\sqrt{3} - 7$

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

(A) $\frac{4}{e^{\pi/4}} - \frac{\pi}{2} - 1$

(B) $\frac{2}{e^{\pi/4}} + \frac{\pi}{4} - 1$

(D) $\frac{2}{e^{\pi/4}} - \frac{\pi}{4} - 1$

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

(A) $\frac{1 - e^3}{e^4}$

(B) $\frac{e^{-1}}{e^4}$

(D) $\frac{2e^{-1}}{e^3}$

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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 $\int_{-3}^{3} y(x) \, dx$ is :

(A) 36

(B) 24

(D) 30

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