GATE Questions & Answers of Differential equations Mechanical Engineering

$ 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 d2ydx2=0 with the bounndary conditiions y=5 at x=0 and dydx=2 at  x = 10, f(15) = _______

The general solution of the differential equation dydx=cosx+y,, with c as a constant, is

Consider two solutions xt=x1t and xt=x2t of the differential equation

d2xtdt2+xt=0,t>0,such that x10=1,dx1tdtt=0=0,x20=0,dx2tdtt=0=1.

The Wronskian Wt=x1tx2tdx1tdtdx2tdt at t=π2  is

The solution of the initial value problem dydx=-2xy; y0=2 is

Laplace transform of cosωt is ss2+ω2 The Laplace transform of e-2tcos4t is

The partial differential equation ut+uux=2u 2xis a

The function f(t) satisfies the differential equation d2fdt2+f=0 and the auxiliary conditions,f(0)=0,dfdt(0)=4 . The Laplace transform of f(t) is given by

The solution to the differential equation d2udx2-kdudx=0 where k is a constant, subjected to the boundary conditions u(0)=0 and u(L)=U, is

The inverse Laplace transform of the function Fs=1ss+1 is given by

Consider the differential equation x2d2ydx2+xdydx-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 dydx=1+y2x The general solution with constant c is

The Blasius equation d3fdη3+f2d2fdη2=0 is a

The Laplace Transform of a function ft=1s2s+1 The f(t) is

The inverse Laplace transform of 1s2+s is

The solution of xdydx+y=x4 with the condition y(1) = 65 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''+2y'+y=0,y0=0,y1=0 what is y0.5?

The partial differential equation 2φx2+2φy2+φx+φy=0 has

The solution of dydx=y2 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