List of top Mathematics Questions on Differential Equations asked in GATE MA

The general solution of $\displaystyle \frac{dy}{dx} = y$ is:

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).

Let \(x(t), y(t), t \in \mathbb{R}\), be two functions satisfying the following system of differential equations:
\[ x'(t) = y(t), \]
\[ y'(t) = x(t), \]
and \(x(0) = \alpha, y(0) = \beta\), where \(\alpha, \beta\) are real numbers.
Which of the following statements is/are correct?

Let \(y(t)\) be the solution of the initial value problem
\[ y'' + 4y = \begin{cases} t, & 0 \le t \le 2, \\ 2, & 2 < t < \infty, \end{cases} \quad \text{and} \quad y(0) = y'(0) = 0. \]
If \(\alpha = y(\pi/2)\), then the value of \(\frac{4}{\pi}\alpha\) is ................................ (rounded off to 2 decimal places).

Consider the ordinary differential equation (ODE) \[ 4 (\ln x) y'' + 3 y' + y = 0, \quad x>1. \] If \(r_1\) and \(r_2\) are the roots of the indicial equation of the above ODE at the regular singular point \(x=1\), then \(|r_1 - r_2|\) is equal to ................. (rounded off to 2 decimal places).

Let \(u(x, t)\) be the solution of the non-homogeneous wave equation \[ \frac{\partial^2 u}{\partial x^2} - \frac{\partial^2 u}{\partial t^2} = \sin x \sin(2t), \quad 0<x<\pi, \ t>0 \] \[ u(x, 0) = 0, \text{ and } \frac{\partial u}{\partial t}(x, 0) = 0, \quad \text{for } 0 \le x \le \pi, \] \[ u(0, t) = 0, \quad u(\pi, t) = 0, \quad \text{for } t \ge 0. \] Then the value of \( u \left( \frac{\pi}{2}, \frac{3\pi}{2} \right) \) is ................. (rounded off to 2 decimal places).

Let \(\Omega\) be the disk \(x^2 + y^2<4\) in \(\mathbb{R}^2\) with boundary \(\partial\Omega\). If \(u(x, y)\) is the solution of the Dirichlet problem \[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0, \quad (x, y) \in \Omega, \] \[ u(x, y) = 1 + 2x^2, \quad (x, y) \in \partial\Omega, \] then the value of \(u(0,1)\) is ..................

For every \(k \in \mathbb{N} \cup \{0\}\), let \(y_k(x)\) be a polynomial of degree k with \(y_k(1) = 5\). Further, let \(y_k(x)\) satisfy the Legendre equation \[ (1-x^2)y'' - 2xy' + k(k+1)y = 0. \] If \[ \frac{1}{2} \sum_{k=1}^{n} \int_{-1}^{1} (y_k(x) - y_{k-1}(x))^2 dx - \sum_{k=1}^{n} \int_{-1}^{1} (y_k(x))^2 dx = 24, \] for some positive integer n, then the value of n is ..................

Let \(\phi\) and \(\psi\) be two linearly independent solutions of the ordinary differential equation \[ y'' + (2 - \cos x) y = 0, \quad x \in \mathbb{R}. \] Let \(\alpha, \beta \in \mathbb{R}\) be such that \(\alpha<\beta\), \(\phi(\alpha) = \phi(\beta) = 0\) and \(\phi(x) \neq 0\) for all \(x \in (\alpha, \beta)\).
Consider the following statements:
P: \(\phi'(\alpha)\phi'(\beta)>0\).
Q: \(\psi(x) \neq 0\) for all \(x \in (\alpha, \beta)\).
Then

Consider the Cauchy problem \[ x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = u; \] \[ u = f(t) \text{ on the initial curve } \Gamma = (t, t); t>0. \] Consider the following statements:
P: If \(f(t) = 2t + 1\), then there exists a unique solution to the Cauchy problem in a neighbourhood of \(\Gamma\).
Q: If \(f(t) = 2t - 1\), then there exist infinitely many solutions to the Cauchy problem in a neighbourhood of \(\Gamma\).
Then

Let \(u(x, t)\) be the solution of \[ \frac{\partial^2 u}{\partial x^2} - \frac{1}{c^2} \frac{\partial^2 u}{\partial t^2} = 0, \quad x \in (-\infty, \infty), t>0, \] \[ u(x, 0) = \sin x, \quad x \in (-\infty, \infty), \] \[ \frac{\partial u}{\partial t}(x, 0) = \cos x, \quad x \in (-\infty, \infty), \] for some positive real number c.
Let the domain of dependence of the solution u at the point P(3,2) be the line segment on the x-axis with end points Q and R.
If the area of the triangle PQR is 8 square units, then the value of \(c^2\) is ................