# GATE Questions & Answers of Differential equations Civil Engineering

#### Differential equations 24 Question(s)

The Laplace transform $F(s)$ of the exponential function, $f(t)=e^{at}$ when  $t\geq0$ , where a is a constant and $(s-a)>0$ , is

##### Show Answer

Consider the following partial differential equation:

$\style{font-family:'Times New Roman'}{3\frac{\partial^2\phi}{\partial x^2}+B\frac{\partial^2\phi}{\partial x\partial y}+3\frac{\partial^2\phi}{\partial y^2}+4\phi=0}$

For this equation to be classified as parabolic , the value of B2 must be _____________.

##### Show Answer

The solution of the equation $\style{font-family:'Times New Roman'}{\frac{dQ}{dt}+Q=1}$ with Q = 0 at t = 0

##### Show Answer

Consider the following second-order differential equation:

$\style{font-family:'Times New Roman'}{y''-4y'+3y=2t-3t^2}$

The particular solution of the differential equation is

##### Show Answer

The type of partial differential equation $\frac{{\partial }^{2}P}{\partial {x}^{2}}+\frac{{\partial }^{2}P}{\partial {y}^{2}}+3\frac{{\partial }^{2}P}{\partial x\partial y}+2\frac{\partial P}{\partial x}-\frac{\partial P}{\partial y}=0\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$ is

##### Show Answer

The solution of the partial differential equation $\frac{\partial u}{\partial t}=\alpha \frac{{\partial }^{2}u}{\partial {x}^{2}}$ is of the form

##### Show Answer

The respective expressions for complimentary function and particular integral part of the solution of the differential equation $\frac{{d}^{4}y}{d{x}^{4}}+3\frac{{d}^{2}y}{d{x}^{2}}=108{x}^{2}$ are

##### Show Answer

Consider the following differential equation:

$\mathrm x(\mathrm{ydx}+\mathrm{xdy})\cos\frac{\mathrm y}{\mathrm x}=\mathrm y(\mathrm{xdy}-\mathrm{ydx})\sin\frac{\mathrm y}{\mathrm x}$

Which of the following is the solution of the above equation (c is an arbitrary constant) ?

##### Show Answer

Consider the following second order linear differential equation

$\frac{{d}^{2}y}{d{x}^{2}}=-12{x}^{2}+24x-20$

The boundary conditions are : at $x=0,y=5$ and at $x=2,y=21$
The value of $y$ at $x=1$ is ___________

##### Show Answer

The integrating factor for the differential equation $\frac{dp}{dt}+{k}_{2}P={k}_{1}{L}_{o}{e}^{-{k}_{1}t}$ is

##### Show Answer

The solution of the ordinary differential equation $\frac{dy}{dx}+2y=0$ for the boundary condition, y = 5 at x = 1 is

##### Show Answer

The solution of the differential equation , with the condition that y=1 at x=1, is

##### Show Answer

The order and degree of the differential equation

$\frac{{d}^{3}y}{d{x}^{3}}+4\sqrt{{\left(\frac{dy}{dx}\right)}^{3}+{y}^{2}}=0$ are respectively

##### Show Answer

The solution to the ordinary differential equation

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

##### Show Answer

The partial differential equation that can be formed from

z = ax + by + ab has the form

##### Show Answer

A parabolic cable is held between two supports at the same level. They horizontal span between the supports is L. The sag at the mid-span is h. The equation of the parabola is y = 4h$\frac{{x}^{2}}{{L}^{2}}\phantom{\rule{0ex}{0ex}}$, where x is the horizontal coordinate and y is the L vertical coordinate with the origin at the centre of the cable. The expression for the total length of the cable is

##### Show Answer

Solution of the differential equation $3y\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}+2x=0$represents a family of

##### Show Answer

Laplace transform for the function $f\left(x\right)=\mathrm{cosh}\left(ax\right)$ is

##### Show Answer

The general solution of $\frac{{d}^{2}y}{d{x}^{2}}+y=0$ is

##### Show Answer

The equation ${k}_{x}\frac{{\partial }^{2}h}{\partial {x}^{2}}+{k}_{z}\frac{{\partial }^{2}h}{\partial {z}^{2}}=0$ can be transformed to $\frac{{\partial }^{2}h}{\partial {{x}_{t}}^{2}}+\frac{{\partial }^{2}h}{\partial {z}^{2}}=0$ by substituting

##### Show Answer

Solution of $\frac{dy}{dx}=-\frac{x}{y}$ at x = 1 and y = $\sqrt{3}$ is

##### Show Answer

The degree of the differential equation $\frac{{\mathit{d}}^{2}x}{\mathit{d}{t}^{2}}+2{x}^{3}=0$ is

##### Show Answer

The solution for the differential equation $\frac{\mathit{d}y}{\mathit{d}x}={x}^{2}y$ with the condition that y = 1 at x = 0 is

##### Show Answer

A body originally at 60°C cools down to 40°C in 15 minutes when kept in air at a temperature of 25°C. What will be the temperature of the body at the end of 30 minutes?