List of top Engineering Mathematics Questions on Differential Equations asked in GATE XE

Let \( y(x) \) be the solution of the initial value problem, \[ x^2 y'' + xy' - y = 0, x>0, y(1) = 0, y'(1) = 2. \] Then the value of \( y'\left(\frac{1}{2}\right) \) is equal to (Answer in integer) ................

Consider the second-order PDE: \[ 4x^2 \frac{\partial^2 u}{\partial x^2} + 4(x+y)\frac{\partial^2 u}{\partial x \partial y} + (x^2+y^2)\frac{\partial^2 u}{\partial y^2} - u=0. \] Which one of the following statements is correct?

Let $y(x)$ be the solution of the initial value problem, \[ x^{2}y'' + xy' - y = 0, x>0, \] with $y(1)=0$, $y'(1)=2$. Then the value of $y'\!\left(\tfrac{1}{2}\right)$ is equal to (Answer in integer):

Let \( y(x) \) be the solution of the initial value problem, \[ x^2 y'' + xy' - y = 0, x>0, y(1) = 0, y'(1) = 2. \] Then the value of \( y'\left(\frac{1}{2}\right) \) is equal to (Answer in integer) ................

If the polynomial \[ P(x)=a_0+a_1x+a_2\,x(x-1)+a_3\,x(x-1)(x-2) \] interpolates the points $(0,2)$, $(1,3)$, $(2,2)$, and $(3,5)$, then the value of $P\!\left(\tfrac{5}{2}\right)$ is ____________ (round off to 2 decimal places).

Let $f:\mathbb{R}^2 \to \mathbb{R}$ be defined by \[ f(x,y)= \begin{cases} \dfrac{xy}{|x|+y}, & y \neq -|x|, \\[6pt] 0, & \text{otherwise}. \end{cases} \] Then which one of the following statements is TRUE?

If the partial differential equation \[ (x + 2) \frac{\partial^2 u}{\partial x^2} + 2(x + y) \frac{\partial^2 u}{\partial x \partial y} + 2(y - 1) \frac{\partial^2 u}{\partial y^2} - 3y^2 \frac{\partial u}{\partial y} = 0 \] is parabolic on the circle \( (x - a)^2 + (y - b)^2 = r^2 \), then the values of \( a \), \( b \), and \( r \) are given by:

Let \( y_1(x) \) and \( y_2(x) \) be two linearly independent solutions of the differential equation: \[ x^2 \frac{d^2y}{dx^2} - 2x \frac{dy}{dx} + 2y = 0, \quad x>0. \] Let \( W(y_1, y_2)(x) \) denote the Wronskian of \( y_1(x) \) and \( y_2(x) \) at \( x \). If \( W(y_1, y_2)(1) = 1 \), then \( W(y_1, y_2)(2) \) is equal to ________.