List of top Mathematics Questions on Differential Equations

Let $ f(x) = x - 1 $ and $ g(x) = e^x $ for $ x \in \mathbb{R} $. If $\frac{dy}{dx} = \left( e^{-2\sqrt{x}} g\left(f\left(f(x)\right)\right) - \frac{y}{\sqrt{x}} \right), \, y(0) = 0,$ then $ y(1) $ is

Consider the system of ordinary differential equations \[ \frac{dX}{dt} = MX, \] where $ M $ is a $ 6 \times 6 $ skew-symmetric matrix with entries in $ \mathbb{R} $. Then, for this system, the origin is a stable critical point for

The partial differential equation \[ (1 + x^2) \frac{\partial^2 u}{\partial x^2} + 2x(1 - y^2) \frac{\partial^2 u}{\partial x \partial y} + (1 - y^2) \frac{\partial^2 u}{\partial y^2} + x \frac{\partial u}{\partial x} + (1 - y^2) \frac{\partial u}{\partial y} = 0 \] is:

Let $ y = P_n(x) $ be the unique polynomial of degree $ n $ satisfying the Legendre differential equation \[ (1 - x^2)y'' - 2xy' + n(n + 1)y = 0 \quad {and} \quad y(1) = 1. \] Then, the value of $ P_{11}'(1) $ is equal to _________ (in integer).

Let $ W $ be the vector space (over $ \mathbb{R} $) consisting of all bounded real-valued solutions of the differential equation \[ \frac{d^4y}{dx^4} + 2 \frac{d^2y}{dx^2} + y = 0. \] Then, the dimension of $ W $ is _________ (in integer).

Let $ y(x) $ be the solution of the differential equation \[ x^2 y'' + 7xy' + 9y = x^{-3} \log_e x, \quad x>0, \] satisfying $ y(1) = 0 $ and $ y'(1) = 0 $. Then, the value of $ y(e) $ is equal to:

Let $ y_1(x) $ and $ y_2(x) $ be the two linearly independent solutions of the differential equation

\[ (1 + x^2) y'' - x y' + (\cos^2 x) y = 0, \]
satisfying the initial conditions

\[ y_1(0) = 3, \quad y_1'(0) = -1, \quad y_2(0) = -5, \quad y_2'(0) = 2. \]
Define

\[ W(x) = \left| \begin{array}{cc} y_1(x) & y_2(x) \\ y_1'(x) & y_2'(x) \\ \end{array} \right|. \]

Then, the value of $ W\left( \frac{1}{2} \right) $ is:

Let $ u(x, t) $ be the solution of the initial value problem \[ \frac{\partial u}{\partial t} + 3 \frac{\partial u}{\partial x} = u, \quad x \in \mathbb{R}, \quad t>0, \quad u(x, 0) = \cos x, \] and let $ v(x, t) $ be the solution of the initial value problem \[ \frac{\partial v}{\partial t} + 3 \frac{\partial v}{\partial x} = v^2, \quad x \in \mathbb{R}, \quad t>0, \quad v(x, 0) = \cos x. \] Then, which of the following is/are TRUE?

Let $ y = P_n(x) $ be the unique polynomial of degree $ n $ satisfying the Legendre differential equation \[ (1 - x^2)y'' - 2xy' + n(n + 1)y = 0 \quad {and} \quad y(1) = 1. \] Then, the value of $ P_{11}'(1) $ is equal to _________ (in integer).

Let $ W $ be the vector space (over $ \mathbb{R} $) consisting of all bounded real-valued solutions of the differential equation \[ \frac{d^4y}{dx^4} + 2 \frac{d^2y}{dx^2} + y = 0. \] Then, the dimension of $ W $ is _________ (in integer).

The degree of differential equation \[ 9 \frac{d^2y}{dx^2} = \left\{ 1 + \left(\frac{dy}{dx}\right)^2 \right\}^{\tfrac{3}{2}} \] is

If $ y = e^{\tan^{-1}x} $, prove that $ (1 + x^2)\frac{d^2y}{dx^2} + (2x - 1)\frac{dy}{dx} = 0 $.

Obtain the differential equation of the family of curves $y = \frac{2ce^{2x}}{1+ce^{2x}}$.

Solve: $(\tan^{-1}y - x)dy = (1+y^2)dx$.

Find a particular solution of the differential equation $ (x - y) (dx + dy) = dx - dy $ when $ y = -1 $ if $ x = 0 $.

Find the general solution of differential equation $ ydx + (x - y^2)dy = 0 $.

For the curve $ \sqrt{x} + \sqrt{y} = 1 $, find the value of $ \frac{dy}{dx} $ at the point $ \left(\frac{1}{9}, \frac{1}{9}\right) $.

The general solution of the differential equation $\left( x \sin \frac{y}{x} \right) \frac{dy}{dx} = y \sin \frac{y}{x} - x$ is
Identify the correct option from the following:

The differential equation corresponding to the family of parabolas whose axis is along $x = 1$ is
Identify the correct option from the following:

The general solution of the differential equation $\frac{dy}{dx} + xy = 4x - 2y + 8$ is