GATE EC Differential Equations questions

The function $y(t)$ satisfies $$t^{2} y^{\prime \prime}(t)-2 t y^{\prime}(t)+2 y(t)=0,$$ where $y^{\prime}(t)$ and $y^{\prime \prime}(t)$ denote the first and second derivatives of $y(t)$, respectively. Given $y^{\prime}(0)=1$ and $y^{\prime}(1)=-1$, the maximum value of $y(t)$ over $[0,1]$ is $\_\_\_\_\_\_\_$ (rounded off to two decimal …

The general form of the complementary function of a differential equation is given by $y(t)=(A t+B) e^{-2 t}$, where $A$ and $B$ are real constants determined by the initial condition. The corresponding differential equation is

Consider the following partial differential equation $\text{(PDE)}$ $$a \frac{ \partial^{2} f(x ,y)}{\partial x^{2}} + b \frac{ \partial^{2} f(x ,y)}{\partial y^{2}} = f(x, y),$$ where $a$ and $b$ are distinct positive real numbers. Select the combination(s) of values of the real parameters $\xi$ and $\eta$ such that …

Consider the following wave equation, $$\frac{\partial^{2}f(x, t)}{\partial t^{2}} = 10000 \frac{\partial^{2}f(x, t)}{\partial x^{2}}$$ Which of the given options is/are solution(s) to the given wave equation?

Consider the differential equation given below. $$\frac{dy}{dx}+\frac{x}{1-x^{2}}y=x\sqrt{y}$$ The integrating factor of the differential equation is

The general solution of $\dfrac{\mathrm{d^{2}} y}{\mathrm{d} x^{2}}-6\dfrac{\mathrm{d} y}{\mathrm{d} x}+9y=0$ is

Which one of the following options contains two solutions of the differential equation $\dfrac{\mathrm{d} y}{\mathrm{d} x}=\left ( y-1 \right )x?$

The families of curves represented by the solution of the equation $$\frac{dy}{dx}=\: – \left(\frac{x}{y} \right)^n$$ for $n=-1$ and $n= +1,$ respectively, are

Consider the homogenous ordinary differential equation $$x^{2}\frac{d^{2}y}{dx^{2}}-3x\frac{dy}{dx}+3y=0, \quad x>0$$ with $y(x)$ as a general solution. Given that $$y(1)=1 \quad \text{and} \quad y(2)=14$$ the value of $y(1.5),$ rounded off to two decimal places, is________.

Let the input be $u$ and the output be $y$ of a system, and the other parameters are real constants. Identify which among the following systems is not a linear system:

A curve passes through the point $\left ( x=1,y=0 \right )$ and satisfies the differential equation $\dfrac{\mathrm{dy} }{\mathrm{d} x}=\dfrac{x^{2}+y^{2}}{2y}+\dfrac{y}{x}.$ The equation that describes the curve is

The position of a particle $y\left ( t \right )$ is described by the differential equation: $$\frac{d^{2}y}{dt^{2}}=-\frac{dy}{dt}-\frac{5y}{4}.$$ The initial conditions are $y\left ( 0 \right )=1$ and $\frac{dy}{dt}\mid_{t=0}=0$. The position (accurate to two decimal places) of the particle at $t=\pi$ is _________.

Which one of the following is the general solution of the first order differential equation $$\frac{dy}{dx}=(x+y-1)^{2},$$ where $x,y$ are real?

The general solution of the differential equation $\frac{d^2y}{dx^2}+2\frac{dy}{dx}-5y=0$ in terms of arbitrary constants $K_1$ and $K_2$ is

The response of the system $G(s)=\frac{s-2}{(s+1)(s+3)}$ to the unit step input $u(t)$ is $y(t)$. The value of $\frac{dy}{dt}$ at $t=0^{+}$ is _________

The ordinary differential equation $$\frac{dx}{dt}=-3x+2, \text{ with }x(0) = 1$$ is to be solved using the forward Euler method. The largest time step that can be used to solve the equation without making the numerical solution unstable is _________

A positive charge $q$ is placed at $x=0$ between two infinte metal plates placed at $x=-d$ and at $x=+d$ respectively. The metal plates lie in the $yz$ plane. The charge is at rest at $t=0$, when a voltage $+V$ is applied to the plate at $-d$ and voltage $-V$ is applied to the plate at $x=+d$. Assume that the quantity of the charge $q$ …

Consider the first order initial value problem $$y’= y+2x-x^2 ,\ y(0)=1,\ (0 \leq x < \infty)$$ with exact solution $y(x) = x^2 + e^x$. For $x = 0.1$, the percentage difference between the exact solution and the solution obtained using a single iteration of the second-order Runga-Kutta method with step-size $h=0.1$ is _______

The particular solution of the initial value problem given below is $$\frac{d^2y}{dx^2}+12\frac{dy}{dx}+36y=0\hspace{0.3cm} \text{ with } \hspace{0.3cm}y(0)=3\hspace{0.3cm} \text{ and }\hspace{0.3cm} \frac{dy}{dx} \bigg| _{x=0} =-36$$

The solution of the differential equation $\frac{d^2y}{dt^2} + 2 \frac{dy}{dt}+y=0$ with $y(0)=y’(0)=1$ is