GATE AE (Aerospace Engineering) Differential Equations previous year questions from the Engineering Mathematics section, with figures and the official answer key where available.
Engineering Mathematics - Differential Equations
The following equation with respect to $\varphi(x,t)$, where $a$ is a non-zero constant, represents ________. $$\frac{\partial \varphi}{\partial t} + a\frac{\partial \varphi}{\partial x} = 0$$
Official answer: A
Engineering Mathematics - Differential Equations
Consider the differential equation with the initial conditions given below. If $y(x)$ is the solution of the equation, the value of the slope, $\dfrac{dy}{dx}$, at $x = \ln(2)$ is ________ (rounded off to three decimal places). $$\frac{d^2y}{dx^2} + 2\frac{dy}{dx} + y = 0 \text{ with } y|_{x=0} = 0 \text{ and } \frac{dy}{dx}\Big|_{x=0} = 1$$
Official answer: 0.151 to 0.155
Engineering Mathematics - Differential Equations
An object of mass 1 kg is launched with an initial speed of $v_o$ into a large tank filled with a viscous liquid. The liquid exerts a resistive force (drag) of the form $D = \alpha v$ on any object that is moving inside it, where $v$ is the instantaneous speed of the object and $\alpha = 1$ kg/s. If the effect of gravity is ignored, the time taken by the object to slow down to the speed $v_o/2$ is ________ s (rounded off to 2 decimal places). Assume that the tank is sufficiently large for the above deceleration to happen inside the tank.
Official answer: 0.67 to 0.71
Engineering Mathematics - Differential Equations
The partial differential equation $\dfrac{\partial^2 u}{\partial x^2} + 4\dfrac{\partial^2 u}{\partial x\,\partial y} + 2\dfrac{\partial^2 u}{\partial y^2} = 0$ is _____.
Official answer: B
Engineering Mathematics - Differential Equations
Consider the ordinary differential equation: $\dfrac{1}{2}\dfrac{dy}{dx} + \dfrac{y}{x} = 1$. If $y = 2/3$ at $x = 1$, then the value of $y$ at $x = 3$ is ______ (rounded off to the nearest integer).
Official answer: 2
Engineering Mathematics - Differential Equations
The acceleration of a body travelling in a straight line is given by $a = -C_1 - C_2 v^2$ where $v$ is the velocity, and $C_1$, $C_2$ are positive constants. Starting with an initial positive velocity $v_o$, the distance travelled by the body before coming to rest for the first time is:
Official answer: A
Engineering Mathematics - Differential Equations
Given $y = e^{px}\sin qx$, where $p$ and $q$ are non-zero real numbers, the value of the differential expression $$\frac{d^2y}{dx^2} - 2p\frac{dy}{dx} + (p^2 + q^2)y$$ is
Official answer: A
Engineering Mathematics - Differential Equations
Consider the one-dimensional wave equation $\dfrac{\partial u}{\partial t} + \dfrac{\partial u}{\partial x} = 0$ for $-\infty < x < \infty$, $t \geq 0$. For an initial condition $u(x,0) = e^{-x^2}$, the solution at $t = 1$ is
Official answer: A
Engineering Mathematics - Differential Equations
Consider the equation $\dfrac{dy}{dx} + ay = \sin\omega x$, where $a$ and $\omega$ are constants. Given $y = 1$ at $x = 0$, select all correct statement(s) from the following as $x \to \infty$.
Official answer: C, D
Engineering Mathematics - Differential Equations
Consider the differential equation $$x^2\frac{d^2y}{dx^2} + 4x\frac{dy}{dx} + 2y = 0 \quad \text{for } x \geq 1$$ with initial conditions $y = 0$, $\dfrac{dy}{dx} = 1$ at $x = 1$. The value of $y$ at $x = 2$ is ____. (round off to two decimal places)
Official answer: 0.24 to 0.26
Engineering Mathematics - Differential Equations
Consider the differential equation $\dfrac{d^2y}{dx^2} - 2\dfrac{dy}{dx} + y = 0$. The boundary conditions are $y = 0$ and $\dfrac{dy}{dx} = 1$ at $x = 0$. Then the value of $y$ at $x = 1/2$ is
Official answer: C
Engineering Mathematics - Differential Equations
Consider the partial differential equation $\dfrac{\partial^2 f}{\partial x^2} + \dfrac{\partial^2 f}{\partial y^2} = 0$ where $x, y$ are real. If $f(x,y) = a(x)b(y)$, where $a(x)$ and $b(y)$ are real functions, which one of the following statements can be true?
Official answer: C
Engineering Mathematics - Differential Equations
The solution of the differential equation $\dfrac{d^2y}{dx^2} + 3\dfrac{dy}{dx} = 0$, given that $y = 0$ and $\dfrac{dy}{dx} = 1$ at $x = 0$ is
Official answer: B
Engineering Mathematics - Differential Equations
The partial differential equation $\dfrac{\partial u}{\partial t} = \alpha\dfrac{\partial^2 u}{\partial x^2}$, where $\alpha$ is a positive constant, is
Official answer: D
Engineering Mathematics - Differential Equations
Consider a second order linear ordinary differential equation $\dfrac{d^2y}{dx^2} - 4\dfrac{dy}{dx} + 4y = 0$, with the boundary conditions $y(0) = 1$; $\dfrac{dy}{dx}\Big|_{x=0} = 1$. The value of $y$ at $x = 1$ is
Official answer: A
Engineering Mathematics - Differential Equations
Given the boundary-value problem $\dfrac{d}{dx}\left(x\dfrac{dy}{dx}\right) + ky = 0,\ 0 < x < 1$, with $y(0) = y(1) = 0$. Then the solutions of the boundary-value problem for $k = 1$ (given by $y_1$) and $k = 5$ (given by $y_5$) satisfy:
Official answer: A
Engineering Mathematics - Differential Equations
Solution to the boundary-value problem $-9\dfrac{d^2u}{dx^2} + u = 5x,\ 0 < x < 3$ with $u(0) = 0,\ \dfrac{du}{dx}\Big|_{x=3} = 0$ is
Official answer: A
Engineering Mathematics - Differential Equations
The Laplace transform $\mathcal{L}(u(t)) = U(s)$, for the solution $u(t)$ of the problem $\dfrac{d^2u}{dt^2} + 2\dfrac{du}{dt} + u = 1,\ t > 0$ with initial conditions $u(0) = 0,\ \dfrac{du(0)}{dt} = 5$ is given by:
Official answer: B