List of top Mathematics Questions on Differential Equations

For the curve \( y = 5x - 2x^3 \), if \( x increases at the rate of 2 units/s, then how fast is the slope of the curve changing when x = 2?

Solve the differential equation: \[ (1 + x^2) \frac{dy}{dx} + 2xy = 4x^2. \]

Let \( \varphi : (0, \infty) \to \mathbb{R} \) be the solution of the differential equation \[ x^2 \frac{d^2 y}{dx^2} - x \frac{dy}{dx} + y = 6x \ln x, \] satisfying \( \varphi(1) = -3 \) and \( \varphi(e) = 0 \). Then, the value of \( \varphi'(1) \) is equal to ............ (rounded off to two decimal places).

Let the subspace \( H \) of \( P_3(\mathbb{R}) \) be defined as \[ H = \{ p(x) \in P_3(\mathbb{R}) : xp'(x) = 3p(x) \}. \] Then, the dimension of \( H \) is equal to ..............

Let \( \varphi : \mathbb{R} \to \mathbb{R} \) be the solution of the differential equation \[ \frac{dy}{dx} + 2xy = 2 + 4x^2, \] satisfying \( \varphi(0) = 0 \). Then, the value of \( \varphi(2) \) is equal to ............. (rounded off to two decimal places).

Let \( \varphi : \mathbb{R} \to \mathbb{R} \) be the solution of the differential equation \[ 4 \frac{d^2 y}{dx^2} + 16 \frac{dy}{dx} + 25y = 0 \] satisfying \( \varphi(0) = 1 \) and \( \varphi'(0) = -\frac{1}{2} \). Then, the value of \( \lim_{x \to \infty} e^{2x} \varphi(x) \) is equal to ............ (rounded off to two decimal places).

Let \( \varphi : (-1, \infty) \to (0, \infty) \) be the solution of the differential equation \[ \frac{dy}{dx} = 2 y e^x = 2 e^x \sqrt{y}, \] satisfying \( \varphi(0) = 1 \). Then, which of the following is/are TRUE?

If \( M, N, \mu, w : \mathbb{R}^2 \to \mathbb{R} \) are differentiable functions with continuous partial derivatives, satisfying \[ \mu(x, y) M(x, y) \, dx + \mu(x, y) N(x, y) \, dy = dw, \] then which one of the following is TRUE?

Let \( \varphi : \mathbb{R} \to \mathbb{R} \) be the solution of the differential equation \[ x \frac{dy}{dx} = (y - 1)(y - 3), \] satisfying \( \varphi(0) = 2 \). Then, which one of the following is TRUE?

Let \( \varphi : (0, \infty) \to \mathbb{R} \) be the solution of the differential equation \[ x \frac{dy}{dx} = \left( \ln y - \ln x \right) y, \] satisfying \( \varphi(1) = e^2 \). Then, the value of \( \varphi(2) \) is equal to:

Which one of the following is the general solution of the differential equation \[ \frac{d^2 y}{dx^2} - 8 \frac{dy}{dx} + 16y = 2e^{4x} ? \]

For which one of the following choices of \( N(x, y) \), is the equation \[ (e^x \sin y - 2y \sin x) \, dx + N(x, y) \, dy = 0 \] an exact differential equation?

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:

The differential equation of the family of circles passing the origin and having center at the line y = x is:

The solution curve of the differential equation \( y \frac{dx}{dy} = x (\log_e x - \log_e y + 1) \), \( x > 0 \), \( y > 0 \) passing through the point \( (e, 1) \) is

Let \( y = y(x) \) be the solution of the differential equation \[ \frac{dy}{dx} = \frac{(\tan x) + y}{\sin x (\sec x - \sin x \tan x)} \], \( x \in \left( 0, \frac{\pi}{2} \right) \) satisfying the condition \( y \left( \frac{\pi}{4} \right) = 2 \). Then, \( y \left( \frac{\pi}{3} \right) \) is

The order and degree of the differential equation: \[ \left[ 1 + \left(\frac{dy}{dx}\right)^2 \right]^3 = \frac{d^2y}{dx^2} \] respectively are:

The order and degree of the differential equation: \[ \left[ 1 + \left( \frac{dy}{dx} \right)^2 \right]^3 = \frac{d^2y}{dx^2}, \] respectively, are:

Let \(f\) be a differential function with
\[ \lim_{x \to \infty} f(x) = 0. \text{ If } y' + y f'(x) - f(x) f'(x) = 0, \lim_{x \to \infty} y(x) = 0 \text{ then,} \]