Recent questions tagged differential-equations

The order and degree of the following differential equation are $m$ and $n$, respectively.\[\frac{\partial^{3} \varphi}{\partial x^{3}}+\frac{\partial^{2} \varphi}{\parti...

Consider the following differential equation\[\frac{\partial y}{\partial x}=3 \frac{\partial y}{\partial t}+y\]If $y(x, 0)=10 e^{-2 x}$, then the solution of the differen...

Let $y$ be the solution of the differential equation with the initial conditions given below. If $y(x=2)=A \ln 2$, then the value of $A$ is ________ (rounded off to $2$ d...

​​​In order to numerically solve the ordinary differential equation $\frac{d y}{d t}=-y$ for $t>0$, with an initial condition $y(0)=1$, the following scheme is employed\[...

If $x(t)$ satisfies the differential equation\[t \frac{d x}{d t}+(t-x)=0\]subject to the condition $x(1)=0$, then the value of $x(2)$ is __________ (rounded off to 2 deci...

Consider the second-order linear ordinary differential equation$$x^2 \frac{d^2 y}{d x^2}+x \frac{d y}{d x}-y=0, \quad x \geq 1$$with the initial conditions$$y(x=1)=6,\lef...

The initial value problem$$\frac{d y}{d t}+2 y=0, \quad y(0)=1$$is solved numerically using the forward Euler's method with a constant and positive time step of $\Delta t...

For the exact differential equation,$$\frac{du}{dx}= \frac{-xu^{2}}{2+x^{2}u},$$which one of the following is the solution?$u^{2} + 2x^{2} =$ constant$xu^{2} + u =$ const...

Solution of $\triangledown ^{2} T = 0$ in a square domain ($0 < x< 1$ and $0 < y< 1$) with boundary conditions: $T \left ( x,0 \right ) = x; T \left ( 0,y \right ) = y; T...

Consider the following differential equation$$\left ( 1+y \right )\frac{dy}{dx}=y.$$The solution of the equation that satisfies condition $y(1)=1$ is$2ye^{y}=e^{x}+e$$y^{...

If $y(x)$ satisfies the differential equation $(\sin x) \dfrac{\mathrm{d}y }{\mathrm{d} x} + y \cos x = 1,$subject to the condition $y(\pi/2) = \pi/2,$ then $y(\pi/6)$ is...

The Dirac-delta function $\left ( \delta \left ( t-t_{0} \right ) \right )$ for $\text{t}$, $t_{0} \in \mathbb{R}$, has the following property$$\int_{a}^{b}\varphi \left ...

The ordinary differential equation $\dfrac{dy}{dt}=-\pi y$ subject to an initial condition $y\left ( 0 \right )=1$ is solved numerically using the following scheme:$$\fra...

The solution of $$\dfrac{d^2y}{dt^2}-y=1,$$ which additionally satisfies $y \bigg \vert_{t=0} = \dfrac{dy}{dt} \bigg \vert_{t=0}=0$ in the Laplace $s$-domain is$\dfrac{1}...

The Laplace transform of a function $f(t)$ is $L( f )=\dfrac{1}{(s^{2}+\omega ^{2})}.$ Then, $f(t)$ is$f\left ( t \right )=\dfrac{1}{\omega ^{2}}\left ( 1-\cos\:\omega t ...

Consider two exponentially distributed random variables $\text{X and Y}$, both having a mean of $0.50$. Let $Z=X+Y$ and $r$ be the correlation between $\text{X and Y}$.If...

A diffferential equation is given as $$x^2 \frac{d^2y}{dx^2} – 2x \frac{dy}{dx} +2y =4$$ The solution of the differential equation in terms of arbitrary constants $C_1$ a...

A harmonic function is analytic if it satisfies the Laplace equation. If $u(x,y)=2x^2-2y^2+4xy$ is a harmonic function, then its conjugate harmonic function $v(x,y)$ is$4...

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

Given the ordinary differential equation $$\dfrac{d^2y}{dx^2}+\dfrac{dy}{dx}-6y=0$$ with $y(0)=0$ and $\dfrac{dy}{dx}(0)=1$, the value of $y(1)$ is __________ (correct to...

If $y$ is the solution of the differential equation $y^3 \dfrac{dy}{dx}+x^3 = 0, \: y(0)=1,$ the value of $y(-1)$ is$-2$$-1$$0$$1$

Consider a function $u$ which depends on position $x$ and time $t$. The partial differential equation $$\frac{\partial u}{\partial t} = \frac{\partial^2 u }{\partial x^2}...

$F(s)$ is the Laplace transform of the function $f(t) =2t^2 e^{-t}$. $F(1)$ is _______ (correct to two decimal places).

An explicit forward Euler method is used to numerically integrate the differential equation $\dfrac{dy}{dt} = y$ using a time step of $0.1$. With the initial condition $y...

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 ________.

The Laplace transform of $te^{t}$ is$\dfrac{s}{(s+1)^{2}} \\$$\dfrac{1}{(s-1)^{2}} \\$$\dfrac{1}{(s+1)^{2}} \\$$\dfrac{s}{(s-1)}$

The differential equation $\dfrac{d^{2}y}{dx^{2}}+16y=0$ for $y(x)$ with the two boundary conditions $\dfrac{dy}{dx}\bigg \vert _{x=0}=1$ and $\dfrac{dy}{dx}\bigg \vert_{...