Differential Equations - General Aptitude - General Aptitude (GA) Previous Year Questions

The minimum possible order of a homogeneous linear ordinary differential equation with real constant coefficients having \(x^2 \sin(x)\) as a solution is equal to ______

The Lagrangian of a system in terms of polar coordinates \((r, \theta)\) is given by \(L = \frac{1}{2} m \dot{r}^2 + \frac{1}{2} m (\dot{r}^2 + r^2 \dot{\theta}^2) - m g r (1 - \cos(\theta))\), where \(m\) is the mass, \(g\) is the acceleration due to gravity and \(\dot{s}\) denotes the derivative of \(s\) with respect to time. Then the equations of motion are

If \(y(x)\) satisfies the initial value problem \((x^2 + y) dx = x dy, \; y(1) = 2\), then \(y(2)\) is equal to ______

Let \(u(x, y) = 2f(y) \cos(x - 2y), (x, y) \in \mathbb{R}^2\), be a solution of the initial value problem \[2u_x + u_y = u\\u(x, 0) = \cos(x).\] Then \(f(1)\) is equal to

Let \(u(x, t), x \in \mathbb{R}, t \ge 0\), be the solution of the initial value problem \[u_{tt} = u_{xx}\\u(x, 0) = x\\u_t(x, 0) = 1.\] Then \(u(2, 2)\) is equal to ________