# GATE Questions & Answers of Differential equations Mechanical Engineering

#### Differential equations 28 Question(s)

$F\left(s\right)$ is the Laplace transform of the function

$F\left(t\right)=2t^2e^{-t}$

$F\left(1\right)$ is ________ (correct to two decimal places).

Consider a function $u$ which depends on position $x$ and time $t$. The partial differential equation

$\frac{\partial u}{\partial t}=\frac{\partial^2u}{\partial x^2}$

is known as the

If y is the solution of the differential equation $y^3\;\frac{dy}{dx}+x^3=0,\;y\left(0\right)=1$ , the value of $y\left(-1\right)$ is

Given the ordinary differential equation

$\frac{d^2y}{dx^2}+\frac{dy}{dx}-6y=0$

with $y\left(0\right)=0$ and $\frac{dy}{dx}\left(0\right)=1$ , the value of $y\left(1\right)$ is _________ (correct to two decimal places).

Consider the following partial differential equation for $\style{font-family:'Times New Roman'}{u(x\mathit,y)}$ with the constant >1:

$\style{font-family:'Times New Roman'}{\frac{\partial u}{\partial y}+C\frac{\partial u}{\partial x}=0}$

Solution of this equation is

The differential equation $\style{font-family:'Times New Roman'}{\frac{d^2y}{dx^2}+16y=0}$ for $\style{font-family:'Times New Roman'}{y\left(x\right)}$ with the two boundary conditions $\style{font-family:'Times New Roman'}{{\left.\frac{dy}{dx}\right|}_{x-0}=1\;\mathrm{and}\;{\left.\frac{dy}{dx}\right|}_{x-\frac{\mathrm\pi}2}=-1}$ has

The Laplace transform of teis

Consider the differential equation 3y(x)+27 y (x)=0 with initial conditions y (0)=0 and y’(0)=2000. The value of y at x =1 is___________

If y = f(x) is the  solution of $\frac{{\mathrm{d}}^{2}\mathrm{y}}{{\mathrm{dx}}^{2}}=0$ with the bounndary conditiions $y=5$ at x=0 and $\frac{\mathrm{dy}}{\mathrm{dx}}=2$ at  x = 10, f(15) = _______

The general solution of the differential equation $\frac{dy}{dx}=\mathrm{cos}\left(x+y\right),$, with c as a constant, is

Consider two solutions $x\left(t\right)={x}_{1}\left(t\right)$ and $x\left(t\right)={x}_{2}\left(t\right)$ of the differential equation

$\frac{{d}^{2}x\left(t\right)}{d{t}^{2}}+x\left(t\right)=0,t>0,$such that ${x}_{1}\left(0\right)=1,{\overline{)\frac{d{x}_{1}\left(t\right)}{dt}}}_{t=0}=0,{x}_{2}\left(0\right)=0,{\overline{)\frac{d{x}_{2}\left(t\right)}{dt}}}_{t=0}=1.$

The Wronskian $W\left(t\right)=\left|\begin{array}{cc}{x}_{1}\left(t\right)& {x}_{2}\left(t\right)\\ \frac{d{x}_{1}\left(t\right)}{dt}& \frac{d{x}_{2}\left(t\right)}{dt}\end{array}\right|\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$ at is

The solution of the initial value problem is

Laplace transform of $cos\left(\omega t\right)$ is $\frac{s}{{s}^{2}+{\omega }^{2}}$ The Laplace transform of ${e}^{-2t}\mathrm{cos}\left(4t\right)$ is

The partial differential equation $\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=\frac{{\partial }^{2}u}{{\partial }^{2}x}$is a

The function $f\left(t\right)$ satisfies the differential equation $\frac{{d}^{2}f}{d{t}^{2}}+f=0$ and the auxiliary conditions,$f\left(0\right)=0,$$\frac{df}{dt}\left(0\right)=4$ . The Laplace transform of $f\left(t\right)$ is given by

The solution to the differential equation $\frac{{d}^{2}u}{d{x}^{2}}-k\frac{du}{dx}=0$ where $k$ is a constant, subjected to the boundary conditions $u\left(0\right)=0$ and $u\left(L\right)=U$, is

The inverse Laplace transform of the function $F\left(s\right)=\frac{1}{s\left(s+1\right)}$ is given by

Consider the differential equation ${x}^{2}\frac{{d}^{2}y}{d{x}^{2}}+x\frac{dy}{dx}-4y=0$ with the boundary conditions of y(0) = 0 and y(1) = 1. The complete solution of the differential equation is

Consider the differential equation $\frac{dy}{dx}=\left(1+{y}^{2}\right)x$ The general solution with constant c is

The Blasius equation $\frac{{d}^{3}f}{d{\eta }^{3}}+\frac{f}{2}\frac{{d}^{2}f}{d{\eta }^{2}}=0$ is a

The Laplace Transform of a function $f\left(t\right)=\frac{1}{{s}^{2}\left(s+1\right)}$ The f(t) is

The inverse Laplace transform of $\frac{1}{\left({s}^{2}+s\right)}$ is

The solution of $x\frac{dy}{dx}+y={x}^{4}$ with the condition y(1) = $\frac{6}{5}$ is

Given that $\ddot x+3x=0,$ and $x\left(0\right)=1,\dot x\left(0\right)=0$, what is x(1)?

It is given that $y\text{'}\text{'}+2y\text{'}+y=0,y\left(0\right)=0,y\left(1\right)=0$ what is $y\left(0.5\right)$?

The partial differential equation $\frac{{\partial }^{2}\phi }{\partial {x}^{2}}+\frac{{\partial }^{2}\phi }{\partial {y}^{2}}+\left(\frac{\partial \phi }{\partial x}\right)+\left(\frac{\partial \phi }{\partial y}\right)=0$ has

The solution of $\frac{dy}{dx}={y}^{2}$ with initial value y(0) = 1 is bounded in the interval
If F(s) is the Laplace transform of function f(t), then Laplace transform of $\int\limits_0^tf\left(\tau\right)\operatorname{d}\tau$ is