Calculus

Differential Equations

Medium

common

The solution of the differential equation $log_e\left(\frac{dy}{dx}\right) = 3x + 4y$ is given by

$4e^{3x} + 3e^{-4y} + C = 0$ , where C is constant of integration

$3e^{3x} + 4e^{-4y} + C = 0$ , where C is constant of integration

$4e^{-3x} + 3e^{4y} + C = 0$ , where C is constant of integration

$3e^{-3x} + 4e^{4y} + C = 0$ , where C is constant of integration

✅ Correct Option: 1

Given $\log_e\left(\frac{dy}{dx}\right) = 3x + 4y$

If $\log_e(A) = B$ , then $A = e^B$

$\frac{dy}{dx} = e^{3x + 4y}$

Using the property $e^{a+b} = e^a \cdot e^b$ :

$\frac{dy}{dx} = e^{3x} \cdot e^{4y}$

Separating the variables:

$\frac{1}{e^{4y}} \, dy = e^{3x} \, dx$

$e^{-4y} \, dy = e^{3x} \, dx$

Integrating both sides:

$\int e^{-4y} \, dy = \int e^{3x} \, dx$

$-\frac{e^{-4y}}{4} = \frac{e^{3x}}{3} + C_1$

Multiplying by $-12$ to clear fractions:

$3e^{-4y} = -4e^{3x} - 12C_1$

$4e^{3x} + 3e^{-4y} = -12C_1$

Since $C_1$ is arbitrary, let $C = -12C_1$ :

$4e^{3x} + 3e^{-4y} = C$

Therefore, the solution is $4e^{3x} + 3e^{-4y} + C = 0$

2026: 19 May Shift 1

Match List-I with List-II

List-I (Differential equation)	List-II (Integrating factor)
(A) $\frac{dy}{dx} + 2y = x$	(I) $e^x$
(B) $\frac{dy}{dx} + \left(\frac{2}{x}\right)y = x^2$	(II) $e^{2x}$
(C) $\frac{dy}{dx} - \left(\frac{2}{x}\right)y = \sin x$	(III) $\frac{1}{x^2}$
(D) $\frac{dy}{dx} + y = e^{-x}$	(IV) $x^2$

Choose the correct answer from the options given below:

2025: 15 May Shift 2

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

2026: 25 May Shift 1

If m and n are the order and the degree of the differential equation :

$x\dfrac{d^2y}{dx^2} = \left[1+\left(\dfrac{dy}{dx}\right)^2\right]^{1/3}$ then which one of the following is TRUE?

2025: 3 June Shift 2

Match List-I with List-II

List-I	List-II
(A) Degree of this differential equation $\frac{d^4y}{dx^4} + 2\log_e\left(\frac{d^3y}{dx^3}\right) = 0$	(I) 1
(B) Order of this differential equation $e^{\left(\frac{dy}{dx}\right)^3} + 3y\left(\frac{d^2y}{dx^2}\right)^3 = 0$	(II) 4
(C) Degree of $\frac{d^4y}{dx^4} + \left(\frac{dy}{dx}\right)^2 = 0$	(III) not defined
(D) Order of the differential equation $2\frac{d^4y}{dx^4} + \left(\frac{d^2y}{dx^2}\right)^5 = 0$	(IV) 2

Choose the correct answer from the options given below:

2026: 19 May Shift 1

The general solution of the given differential equation $\frac{dy}{dx} = \frac{3e^{2x}+3e^{4x}}{e^{x}+e^{-x}}$ (where C is an arbitrary constant)

2025: 3 June Shift 2

For the differential equation $x\frac{dy}{dx} + 2y = x^2\log_e x$

(A) Integrating factor is $2x$

(B) Integrating factor is $x^2$

(D) General Solution is $y = \frac{x^4}{16}(4\log_e|x| - 1) + C$ Where C is an arbitrary constant.

Choose the correct answer from the options given below:

The solution of the differential equation loge(dydx)=3x+4ylog_e\left(\frac{dy}{dx}\right) = 3x + 4yloge​(dxdy​)=3x+4y is given by

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

The solution of the differential equation $log_e\left(\frac{dy}{dx}\right) = 3x + 4y$ is given by