# GATE Questions & Answers of Differential equations Electronics and Communication Engg

#### Differential equations 23 Question(s)

Let (X1, X2) be independent random variables. X1 has mean 0 and variance 1, while X2 has mean 1 and variance 4. The mutual information (X1; X2) between X1 and X2 in bits is_________.

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

The generation solution of the differential equation $\frac{{d}^{2}y}{d{x}^{2}}+2\frac{dy}{dx}-5y=00$ in terms of arbitrary constant K1 and K2 is

Which one of the following is a property of the solutions to the Laplace equation: $\nabla^2f=0?$

The ordinary differential equation

$\frac{dx}{dt}=-3x+2,$ with $\mathrm x\left(0\right)=1$

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 ________

The particular solution of the initial value problem given below is

$\frac{d^2y}{dx^2}+12\frac{dy}{dx}+36y=0\;\;$ with $y\left(0\right)=3$ and $\frac{dy}{dx}\vert_{x=0}=-36$

The solution of the differential equation $\frac{{d}^{2}y}{d{t}^{2}}+2\frac{dy}{dt}+y=0$ with $y(0)=y'(0)=1$ is

The general solution of the differential equation $\frac{dy}{dx}=\frac{1+\mathrm{cos}2y}{1-\mathrm{cos}2x}$ is

Consider the differential equation

$\frac{{d}^{2}x\left(t\right)}{d{t}^{2}}+3\frac{dx\left(t\right)}{dt}+2x\left(t\right)=0$

Given x(0) = 20 and x(1) = 10/e, where e = 2.718, the value of x(2) is _____.

If the characteristic equation of the differential equation

$\frac{{d}^{2}y}{d{x}^{2}}+2\alpha \frac{dy}{dx}+y=0$

has two equal roots, then the values of $\alpha$ are

Which ONE of the following is a linear non-homogeneous differential equation, where x and y are the independent and dependent variables respectively?

If a and b are constants, the most general solution of the differential equation

$\frac{{d}^{2}x}{d{t}^{2}}+2\frac{dx}{dt}+x=0$ is

With initial values y(0)=y′(0)=1, the solution of the differential equation

$\frac{{d}^{2}y}{d{x}^{2}}+4\frac{dy}{dx}+4y=0$

at x=1 is _____.

A system described by a linear, constant coefficient, ordinary, first order differential equation has an exact solution given by y(t) for t > 0 , when the forcing function is x(t) and the initial condition is y(0) . If one wishes to modify the system so that the solution becomes -2y(t) for t > 0 , we need to

With initial condition $x\left(1\right)=0.5$ , the solution of the differential equation,

$t\frac{dx}{dt}+x=t$ is

Consider the differential equation

$\frac{{d}^{2}y\left(t\right)}{d{t}^{2}}+2\frac{dy\left(t\right)}{dt}+y\left(t\right)=\delta \left(t\right)$ with ${\overline{)y\left(t\right)}}_{t={0}^{-}}=-2$ and ${\overline{)\frac{dy}{dt}}}_{t={0}^{-}}=0$

The numerical value of ${\overline{)\frac{dy}{dt}}}_{t={0}^{+}}$ is

 (A) –2 (B) –1 (C) 0 (D) 1

The solution of the differential equation

A function n(x) satisfied the differential equation $\frac{{d}^{2}n\left(x\right)}{d{x}^{2}}-\frac{n\left(x\right)}{{L}^{2}}=0$ where L is a constant. The boundary conditions are: n(0)=K and n ( ∞ ) = 0. The solution to this equation is

Consider differential equation $\frac{dy\left(x\right)}{dx}-y\left(x\right)=x$ with the initial condition y(0) = 0. Using Euler’s first order method with a step size of 0.1, the value of y (0.3) is

The order of the differential equation $\frac{{d}^{2}\mathrm{y}}{d{\mathrm{t}}^{2}}+{\left(\frac{d\mathrm{y}}{d\mathrm{t}}\right)}^{3}+{\mathrm{y}}^{4}={\mathrm{e}}^{-\mathrm{t}}$ is

Match each differential equation in Group I to its family of solution curves from Group II.

 Group I Group II P. $\frac{dy}{dx}=\frac{y}{x}$ 1. Circles Q. $\frac{dy}{dx}=-\frac{y}{x}$ 2. Straight lines R. $\frac{dy}{dx}=\frac{x}{y}$ 3. Hyperbolas S. $\frac{dy}{dx}=-\frac{x}{y}$

Which of the following is a solution to the differential equation $\frac{dx\left(t\right)}{dt}+3x\left(t\right)=0$ ?

The solution of the differential equation ${k}^{2}\frac{{d}^{2}y}{d{x}^{2}}=y-{y}_{2}$ under the boundary conditions (i) y = y1 at x = 0 and (ii) y = y2 at x = $\infty$, where k, y1 and y2 are constants, is