List of top Mathematics Questions on Differential Equations asked in CUET (UG)

Consider the differential equation, $x \frac{dy}{dx} = y(\log_e y - \log_e x + 1)$, then which of the following are true?
(A) It is a linear differential equation
(B) It is a homogenous differential equation
(C) Its general solution is $\log_e(\frac{y}{x}) = Cx$, where C is constant of integration
(D) Its general solution is $\log_e(\frac{x}{y}) = Cy$, where C is constant of integration
(E) If y(1) = 1, then its particular solution is y = x
Choose the correct answer from the options given below:

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 \(x(e^{2y} - 1)dy + (x^2 - 1)e^y dx = 0\) and passes through the point (1, 1). The value of \((\alpha + \beta + \gamma + \delta - C)\) is

Consider the differential equation, $x \frac{dy}{dx} = y(\log_e y - \log_e x + 1)$, then which of the following are true?
(A) It is a linear differential equation
(B) It is a homogenous differential equation
(C) Its general solution is $\log_e(\frac{y}{x}) = Cx$, where C is constant of integration
(D) Its general solution is $\log_e(\frac{x}{y}) = Cy$, where C is constant of integration
(E) If y(1) = 1, then its particular solution is y = x
Choose the correct answer from the options given below:

If \(e^y = \log x\), then which of the following is true?

Which of the following are linear first order differential equations?
(A) $\frac{dy}{dx} + P(x)y = Q(x)$
(B) $\frac{dx}{dy} + P(y)x = Q(y)$
(C) $(x - y)\frac{dy}{dx} = x + 2y$
(D) $(1 + x^2)\frac{dy}{dx} + 2xy = 2$
Choose the correct answer from the options given below:

If \(e^y = \log x\), then which of the following is true?

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

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 \(x(e^{2y} - 1)dy + (x^2 - 1)e^y dx = 0\) and passes through the point (1, 1). The value of \((\alpha + \beta + \gamma + \delta - C)\) is

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

Which of the following are linear first order differential equations?
(A) $\frac{dy}{dx} + P(x)y = Q(x)$
(B) $\frac{dx}{dy} + P(y)x = Q(y)$
(C) $(x - y)\frac{dy}{dx} = x + 2y$
(D) $(1 + x^2)\frac{dy}{dx} + 2xy = 2$
Choose the correct answer from the options given below:

If \[ y = \frac{1}{\sqrt{1 - 4 \sin^2 x \cos^2 x}}, \] then $\frac{dy}{dx}$ is:

If \( x = at^4 \) and \( y = 2at^2 \), then \( \frac{d^2y}{dx^2} \) is equal to:

The particular solution of the differential equation \((y - x^2) dy = (1 - x^3) dx\) with \(y(0) = 1\), is:

The integrating factor of the differential equation \[ (y \log_e y) \frac{dx}{dy} + x = 2 \log_e y \] is:

The degree and order of the differential equation \[ \left( \frac{d^2 y}{dx^2} \right)^{\frac{4}{5}} = 10 \frac{dy}{dx} + 2 \] are:

If the solution of the differential equation \[ \frac{dy}{dx} = \frac{ax + 3}{2y + 5} \] represents a circle, then $a$ is equal to:

For the differential equation \((x \log x) \, dy = (\log x - y) \, dx\):
(A) Degree of the given differential equation is 1.
(B) It is a homogeneous differential equation.
(C) Solution is \(2y \log x + A = (\log x)^2\), where \(A\) is an arbitrary constant.
(D) Solution is \(2y \log x + A = \llog(\ln x)\), where \(A\) is an arbitrary constant.