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

If the radius of curvature (\(\rho\)) at (0, 1) of \(y = e^x\) is \(\alpha\sqrt{\beta}\), then \(\alpha^2+\beta\) is:

The function \(y = \tan^{-1}x\) satisfies differential equation

Which of the following statements are true?
(A) For f(x) = |x|, for all x in [-1, 2]; Lagrange's mean value theorem is satisfied
(B) For f(x) = cosx, for all x in [0, \(\pi\)/2]; Lagrange's mean value theorem is satisfied
(C) For f(x) = \( \frac{1}{x} \), for all x in [-1, 2]; Lagrange's mean value theorem is satisfied
(D) For f(x) = x(x-1)(x-2), for all x in [0, 1/2]; Lagrange's mean value theorem is satisfied
(E) For f(x) = \(x^{1/3}\), for all x in [-1, 1]; Lagrange's mean value theorem is satisfied
Choose the correct answer from the options given below:

Match List-I with List-II

\[ \begin{array}{|l|l|} \hline \textbf{List-I (Functions)} & \textbf{List-II (Concavity and Convexity)} \\ \hline (A) \; f(x) = e^{-x^2} & (I) \; \text{Concave downward in } (-\infty, -1) \\ (B) \; f(x) = (1+x^2)e^{-x} & (II) \; \text{Concave upward in } (-\infty, 1) \\ (C) \; f(x) = 3x^4+4x^3-6x^2+12x+12 & (III) \; \text{Concave downward in } \left(-\tfrac{1}{\sqrt{2}}, \tfrac{1}{\sqrt{2}}\right) \\ (D) \; f(x) = (x+1)^{1/3} & (IV) \; \text{Concave upward in } (-\infty, -1) \\ \hline \end{array} \]
Choose the correct answer from the options given below:

Which one of the following statement is not correct?

The general solution of differential equation \( \frac{d^2y}{dx^2} + 9y = \cos(3x) \) is:

Match List-I with List-II

\[ \begin{array}{|l|l|} \hline \textbf{List-I (Curve)} & \textbf{List-II (Orthogonal trajectory)} \\ \hline (A) \; xy = c & (I) \; \tfrac{y^2}{2} + x^2 = c \\ (B) \; e^x + e^{-y} = c & (II) \; y(y^2 + 3x^2) = c \\ (C) \; y^2 = cx & (III) \; y^2 - x^2 = 2c \\ (D) \; x^2 - y^2 = cx & (IV) \; e^y - e^{-x} = c \\ \hline \end{array} \]
Choose the correct answer from the options given below:

The particular integral of differential equation \( \frac{d^2y}{dx^2} + 2\frac{dy}{dx} + y = e^{-x}\log x \) is:

The solution of the differential equation \( \frac{xdy - ydx}{xdx + ydy} = \sqrt{x^2+y^2} \) is:

The solution of the differential equation \( (xy^3 + y)dx + (2x^2y^2 + 2x + 2y^4)dy = 0 \) is:

The solution of the differential equation \( (x^2 - 4xy - 2y^2)dx + (y^2 - 4xy - 2x^2)dy = 0 \), is

If \( x \in \mathbb{R} \) and a particular integral (P.I.) of \( (D^2 - 2D + 4)y = e^x \sin x \) is \( \frac{1}{2} e^x f(x) \), then \( f(x) \) is:

The orthogonal trajectory of the cardioid \( r = a(1 - \cos\theta) \), where 'a' is an arbitrary constant is:

A complete solution of \( y'' + a_1 y' + a_2 y = 0 \) is \( y = b_1 e^{-x} + b_2 e^{-3x} \), where \( a_1, a_2, b_1 \) and \( b_2 \) are constants, then the respective values of \( a_1 \) and \( a_2 \) are:

The solution of the differential equation \( xdy - ydx = (x^2+y^2)dx \), is