Calculus

Differential Equations

Medium

applied

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

$x e^y + x^2 = C$ , where C is constant of integration

$x e^y + y^2 = C$ , where C is constant of integration

$y e^x + x^2 = C$ , where C is constant of integration

$y e^y + x^2 = C$ , where C is constant of integration

✅ Correct Option: 3

Given differential equation:

$e^x dy + (y e^x + 2x)dx = 0$

Rewriting by separating the terms:

$e^x dy + y e^x dx + 2x dx = 0$

The first two terms $e^x dy + y e^x dx$ form the differential of a product.

The differential of $ye^x$ is:

$d(ye^x) = y \cdot d(e^x) + e^x \cdot dy$

$d(ye^x) = y \cdot e^x dx + e^x dy$

Therefore:

$e^x dy + y e^x dx = d(ye^x)$

The equation becomes:

$d(ye^x) + 2x dx = 0$

Integrating both sides:

$\int d(ye^x) + \int 2x dx = \int 0$

$ye^x + x^2 = C$

Therefore, the general solution is:

$ye^x + x^2 = C$

where $C$ is the constant of integration.

2022: 4 Aug Shift 1

The sum of order and degree of differential equation $2x^2 \cdot \left(\frac{d^2y}{dx^2}\right) - 3 \cdot \left(\frac{dy}{dx}\right)^3 + y = 0$ is

2026: 29 May Shift 1

Match List-I with List-II

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

Choose the correct answer from the options given below:

2023: 22 May Shift 3

The differential equation of the family of curves $y = a \sin(bx + c)$ , a and c are parameters, is :

2026: 21 May Shift 2

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

2025: 30 May Shift 2

The sum of order and degree of the differential equation $y = x\frac{dy}{dx} + 2\sqrt{1 + \left(\frac{dy}{dx}\right)^2}$ is

2022: 4 Aug Shift 1

Solution of the differential equation $\frac{dy}{dx} = x + xy - (1 + y)$ is:

The general solution of the differential equation exdy+(yex+2x)dx=0e^x dy + (y e^x + 2x)dx = 0exdy+(yex+2x)dx=0 is