Calculus > Differential Equations

Hard

applied

Let $e^{\alpha y} + e^{\beta y} + \gamma x^2 + \delta \log|x| + C = 0$, where $C \in \mathbb{R}$ be a particular solution of the differential equation $x(e^{2y} - 1)dy + (x^2 - 1)e^ydx = 0$ and passes through the point $(1, 1)$. The value of $(\alpha + \beta + \gamma + \delta - C)$ is

$e - 1$

$e^2 - 1$

$e + \frac{1}{e}$

$\frac{1}{e}$

✅ Correct Option: 3

Related questions:

2022: 10 Aug Shift 1

General solution of the differential equation $\frac{dy}{dx} + \frac{\sqrt{1 - y^2}}{\sqrt{1 - x^2}} = 0$ is

A. $\tan^{-1} x + \tan^{-1} y = C$

B. $\sin^{-1} x - \cos^{-1} y = C$

C. $x\sqrt{1 - y^2} - y\sqrt{1 - x^2} = C$

D. $\sin^{-1} x + \sin^{-1} y = C$

E. $\cos^{-1} x + \cos^{-1} y = C$

(where C is arbitrary constant)

Choose the correct answer from the options given below

2025: 22 May Shift 1

Match List-I with List-II

List-I	List-II
Differential Equation	Degree
(A) $xy\frac{d^2y}{dx^2} + x\left(\frac{dy}{dx}\right)^2 - y\frac{dy}{dx} = 0$	(I) 3
(B) $\frac{d^2y}{dx^2} + \log\left(\frac{dy}{dx}\right) = 0$	(II) 1
(C) $\left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^3 + \frac{dy}{dx} + 1 = 0$	(III) not defined
(D) $2x^2\left(\frac{d^2y}{dx^2}\right)^3 - 5\left(\frac{dy}{dx}\right)^3 + y = 0$	(IV) 2

Choose the correct answer from the options given below:

2026: 12th May Shift 2

Which of the following differential equations has $y=x$ as one of the particular solution?

2025: 14 May Shift 2

Particular solution of the differential equation $x(1 + y^2)dx - y(1 + x^2)dy = 0$ , given $y = 0$ when $x = 1$ , is

2022: 30 Aug Shift 1

The integrating factor of the differential equation $\cos x \frac{dy}{dx} + y\sin x = 1$ is

2026: 25th May Shift 1

General solution of the differential equation $(e^x + e^{-x})dy = (3e^{2x} + 3e^{4x})dx$ is:

[Where C is an arbitrary constant]

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