Differential Equations (CE)

Pick the CORRECT solution for the following differential equation $\frac{d y}{d x}=e^{x-y}$

The "order" of the following ordinary differential equation is $\qquad$ . $$ \frac{d^3 y}{d x^3}+\left(\frac{d^2 y}{d x^2}\right)^6+\left(\frac{d y}{d x}\right)^4+y=0 $$

Let $y$ be the solution of the initial value problem $y^{\prime}+0.8 y+0.16 y=0$ where $y(0)=3$ and $y^{\prime}(0)=4.5$. Then, $y(1)$ is equal to__________ (rounded off to 1 decimal place).

Which of the following equations belong/belongs to the class of second-order, linear, homogeneous partial differential equations:

For the following partial differential equation, $x \frac{\partial^2 f}{\partial x^2} + y \frac{\partial^2 f}{\partial y^2} = \frac{x^2 + y^2}{2}$ which of the following option(s) is/are CORRECT?

A partial differential equation $$\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0$$ is defined for the two-dimensional field $T: T(x, y)$, inside a planar square domain of size 2 m × 2 m. Three...

A 2 m × 2 m tank of 3 m height has inflow, outflow and stirring mechanisms. Initially, the tank was half-filled with fresh water. At $ t = 0 $, an inflow of a salt solution of concentration 5 g/ $ m^3 $ at the rate of 2...

The second-order differential equation in an unknown function $$u : u(x, y)$$ is defined as $$\frac{\partial^2 u}{\partial x^2}= 2$$ Assuming $$g : g(x)$$, $$f : f(y)$$, and $$h : h(y)$$, the general solution of the abov...

Consider two Ordinary Differential Equations (ODEs): P: $ \dfrac{dy}{dx} = \dfrac{x^4 + 3x^2 y^2 + 2y^4}{x^3 y} $ Q: $ \dfrac{dy}{dx} = -\dfrac{y^2}{x^2} $ Which one of the following options is CORRECT?

The steady-state temperature distribution in a square plate ABCD is governed by the 2-dimensional Laplace equation. The side AB is kept at a temperature of 100°C and the other three sides are kept at a temperature of 0°C...

The solution of the differential equation $\rm \frac{d^3y}{dx^3}-5.5\frac{d^2y}{dx^2}+9.5\frac{dy}{dx}-5y=0$ is expressed as 𝑦 = 𝐶 1 𝑒 2.5𝑥 + 𝐶 2 𝑒 𝛼𝑥 + 𝐶 3 𝑒 𝛽𝑥 , where 𝐶 1 , 𝐶 2 , 𝐶 3 , 𝛼, and 𝛽 are co...

In the differential equation $\frac{dy}{dx}+\alpha\ x\ y =0, \alpha$ is a positive constant. If y = 1.0 at x = 0.0, and y = 0.8 at x = 1.0, the value of α is ________ (rounded off to three decimal places).

A 5 cm long metal rod AB was initially at a uniform temperature of T 0 °C. Thereafter, temperature at both the ends are maintained at 0°C. Neglecting the heat transfer from the lateral surface of the rod, the heat transf...

The function f(x, y) satisfies the Laplace equation $$\Delta$$ 2 f(x, y) = 0 on a circular domain of radius r = 1 with its center at point P with coordinates x = 0, y = 0. The value of this function on the circular bound...

For the equation $${{{d^3}y} \over {d{x^3}}} + x{\left( {{{dy} \over {dx}}} \right)^{3/2}} + {x^2}y = 0$$ the correct description is

Consider the following expression: z = sin(y + it) + cos(y $$-$$ it) where z, y, and t are variables, and $$i = \sqrt { - 1} $$ is a complex number. The partial differential equation derived from the above expression is

Consider the following partial differential equation: $$\,\,3{{{\partial ^2}\phi } \over {\partial {x^2}}} + B{{{\partial ^2}\phi } \over {\partial x\partial y}} + 3{{{\partial ^2}\phi } \over {\partial {y^2}}} + 4\phi =...

Consider the following second $$-$$order differential equation : $$\,y''\,\, - 4y' + 3y = 2t - 3{t^2}\,\,\,$$ The particular solution of the differential equation is

The solution of the equation $$\,{{dQ} \over {dt}} + Q = 1$$ with $$Q=0$$ at $$t=0$$ is

The type of partial differential equation $${{{\partial ^2}p} \over {\partial {x^2}}} + {{{\partial ^2}p} \over {\partial {y^2}}} + 3{{{\partial ^2}p} \over {\partial x\partial y}} + 2{{\partial p} \over {\partial x}} -...

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

Consider the following differential equation $$x\left( {y\,dx + x\,dy} \right)\cos \left( {{y \over x}} \right)$$ $$\,\,\,\,\,\,\,\,\,\, = y\left( {x\,dy - y\,dx} \right)\sin \left( {{y \over x}} \right)$$ Which of the f...

Consider the following second order linear differential equation $${{{d^2}y} \over {d{x^2}}} = - 12{x^2} + 24x - 20$$ The boundary conditions are: at $$x=0, y=5$$ and at $$x=2, y=21$$ The value of $$y$$ at $$x=1$$ is

Water is following at a steady rate through a homogeneous and saturated horizontal soil strip of $$10$$m length. The strip is being subjected to a constant water head $$(H)$$ of $$5$$m at the beginning and $$1$$m at the...

The integrating factor for the differential equation $${{dP} \over {dt}} + {k_2}\,P = {k_1}{L_0}{e^{ - {k_1}t}}\,\,$$ is

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

If $$F\left( s \right) = L\left\{ {f\left( t \right)} \right\} = {{2\left( {s + 1} \right)} \over {{s^2} + 4s + 7}}$$ then the initial and final values of $$f(t)$$ are respectively

The solution of the differential equation $${{dy} \over {dx}} + {y \over x} = x$$ with the condition that $$y=1$$ at $$x=1$$ is

The order and degree of a differential equation $${{{d^3}y} \over {d{x^3}}} + 4\sqrt {{{\left( {{{dy} \over {dx}}} \right)}^3} + {y^2}} = 0$$ are respectively

The solution to the ordinary differential equation $${{{d^2}y} \over {d{x^2}}} + {{dy} \over {dx}} - 6y = 0\,\,\,$$ is

The partial differential equation that can be formed from $$z=ax+by+ab$$ has the form $$\,\,\left( {p = {{\partial z} \over {\partial x}},q = {{\partial z} \over {\partial y}}} \right)\,\,$$

Solution of the differential equation $$3y{{dy} \over {dx}} + 2x = 0$$ represents a family of

Laplace transform of $$f\left( x \right) = \cos \,h\left( {ax} \right)$$ is

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

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

A body originally at $${60^ \circ }$$ cools down to $$40$$ in $$15$$ minutes when kept in air at a temperature of $${25^ \circ }$$c. What will be the temperature of the body at the and of $$30$$ minutes?

The solution of the differential equation $$\,{x^2}{{dy} \over {dx}} + 2xy - x + 1 = 0\,\,\,$$ given that at $$x=1,$$ $$y=0$$ is

The solution $${{{d^2}y} \over {d{x^2}}} + 2{{dy} \over {dx}} + 17y = 0;$$ $$y\left( 0 \right) = 1,{\left( {{{d\,y} \over {d\,x}}} \right)_{x = {\raise0.5ex\hbox{$\scriptstyle \pi $} \kern-0.1em/\kern-0.15em \lower0.25ex...

Laplace transform of $$f\left( t \right) = \cos \left( {pt + q} \right)$$ is

Transformation to linear form by substituting $$v = {y^{1 - n}}$$ of the equation $${{dy} \over {dt}} + p\left( t \right)y = q\left( t \right){y^n},\,\,n > 0$$ will be

The Laplace transform of a function $$f(t)$$ is $$$F\left( s \right) = {{5{s^2} + 23s + 6} \over {s\left( {{s^2} + 2s + 2} \right)}}$$$ As $$t \to \propto ,\,\,f\left( t \right)$$ approaches

Biotransformation of an organic compound having concentration $$(x)$$ can be modeled using an ordinary differential equation $$\,{{d\,x} \over {dt}} + k\,{x^2} = 0,$$ where $$k$$ is the reaction rate constant. If $$x=a$$...

A delayed unit step function is defined as $$$u\left( {t - a} \right) = \left\{ {\matrix{ {0,} & {t < a} \cr {1,} & {t \ge a} \cr } } \right.$$$ Its Laplace transform is ____________.

If $$L$$ denotes the laplace transform of a function, $$L\left\{ {\sin \,\,at} \right\}$$ will be equal to

Using Laplace transforms, solve $${a \over {{s^2} - {a^2}}}\,\,\left( {{d^2}y/d{t^2}} \right) + 4y = 12t\,\,$$ given that $$y=0$$ and $$dy/dt=9$$ at $$t=0$$

The solution for the following differential equation with boundary conditions $$y(0)=2$$ and $$\,\,{y^1}\left( 1 \right) = - 3$$ is where $${{{d^2}y} \over {d{x^2}}} = 3x - 2$$

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

The number of boundary conditions required to solve the differential equation $$\,\,{{{\partial ^2}\phi } \over {\partial {x^2}}} + {{{\partial ^2}\phi } \over {\partial {y^2}}} = 0\,\,$$ is

Let $$F\left( s \right) = L\left[ {f\left( t \right)} \right]$$ denote the Laplace transform of the function $$f(t)$$. Which of the following statements is correct?

The Laplace transform of the function $$\eqalign{ & f\left( t \right) = k,\,0 < t < c \cr & \,\,\,\,\,\,\,\,\, = 0,\,c < t < \infty ,\,\, \cr} $$ is

If $$c$$ is a constant, then the solution of $${{dy} \over {dx}} = 1 + {y^2}$$ is

The Laplace Transform of a unit step function $${u_a}\left( t \right),$$ defined as $$\matrix{ {{u_a}\left( t \right) = 0} & {for\,\,\,t < a\,} \cr { = 1} & {for\,\,\,t > a,} \cr } $$ is

Solve $${{{d^4}y} \over {d{x^4}}} - y = 15\,\cos \,\,2x$$

$${\left( {s + 1} \right)^{ - 2}}$$ is laplace transform of

For the differential equation $$f\left( {x,y} \right){{dy} \over {dx}} + g\left( {x,y} \right) = 0\,\,$$ to be exact is

The differential equation $${{dy} \over {dx}} + py = Q,$$ is a linear equation of first order only if,

Solve $${{{d^4}v} \over {d{x^4}}} + 4{\lambda ^4}v = 1 + x + {x^2}$$

Using Laplace transform, solve the initial value problem $$9{y^{11}} - 6{y^1} + y = 0$$ $$y\left( 0 \right) = 3$$ and $${y^1}\left( 0 \right) = 1,$$ where prime denotes derivative with respect to $$t.$$

The inverse Laplace transform of $${{\left( {s + 9} \right)} \over {\left( {{s^2} + 6s + 13} \right)}}$$ is

The differential equation $${y^{11}} + {\left( {{x^3}\,\sin x} \right)^5}{y^1} + y = \cos {x^3}\,\,\,\,$$ is

The solution of a differential equation $${y^{11}} + 3{y^1} + 2y = 0$$ is of the form

The necessary & sufficient condition for the differential equation of the form $$\,\,M\left( {x,y} \right)dx + N\left( {x,y} \right)dy = 0\,\,$$ to be exact is

The differential equation $${{{d^4}y} \over {d{x^4}}} + P{{{d^2}y} \over {d{x^2}}} + ky = 0\,\,$$ is