Recent questions and answers in Differential Equations

Consider the following differential equation:\[t^{2} \frac{d^{2} y}{d t^{2}}+7 t \frac{d y}{d t}+8 t y=10 \sin (t)\]Which one of the following options is correct?It is a ...

Consider the second-order differential equation\[\frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}+y=0\]with initial conditions\[y(0)=1,\left.\quad \frac{d y}{d x}\right|_{x=0}=1\]...

Consider ordinary differential equations given by$\begin{array}{l} \dot{x}_{1}(t)=2 x_{2}(t) \\ \dot{x}_{2}(t)=r(t) \end{array}$with initial conditions $x_{1}(0)=1$ and $...

Which of the following differential equations is/are nonlinear?$t x(t)+\frac{d x(t)}{d t}=t^{2} e^{t}, \quad x(0)=0$$\frac{1}{2} e^{t}+x(t) \frac{d x(t)}{d t}=0, \quad x(...

In the following differential equation, the numerically obtained value of $y(t),$ at $t=1,$ is ___________ (Round off to $2$ decimal places). $\frac{d y}{d t}=\frac{e^{-...

Consider the initial value problem below. The value of y at $x=\ln{2}$, (rounded off to $3$ decimal places) is ______________.$$\frac{\mathrm{d} y}{\mathrm{d} x}=2x-y,\:\...

The partial differential equation $\frac{\partial^{2}u}{\partial t^{2}}- C^{2} \bigg( \frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}} \bigg )=0;...

Consider the differential equation $\dfrac{d^2y(t)}{dt^2} + 2 \dfrac{dy(t)}{dt} + y(t)=\delta (t)$ with $y(t) \bigg \vert_{t=0^-}= -2$ and $\dfrac{dy}{dt} \bigg \vert _{t...

With initial condition $x(1)=0.5$, the solution of the differential equation $t\dfrac{dx}{dt}+x=t$ is$x=t-\dfrac{1}{2} \\ $$x=t^2-\dfrac{1}{2} \\ $$x=\dfrac{t^2}{2} \\$$x...

Consider a system governed by the following equations $$ \frac{dx_1(t)}{dt} = x_2(t)-x_1(t) \\ \frac{dx_2(t)}{dt} = x_1(t)-x_2(t)$$ The initial conditions are such that $...

Consider the differential equation $(t^{2}-81)\frac{dy}{dt}+5t y=\sin(t)$ with $y(1)=2 \pi$. There exists a unique solution for this differential equation when $t$ belong...

Let a causal LTI system be characterised by the following differential equation, with initial rest condition$\frac{d^{2}y}{dt^{2}}+7\frac{dy}{dt}+10y (t)=4x(t)+5\frac{dx(...

Consider the differential equation $x^2\dfrac{d^2y}{dx^2}+x\dfrac{dy}{dx}-y=0$. Which of the following is a solution to this differential equation for $x>0$?$e^x$$x^2$$1/...

The solution for the differential equation $\dfrac{d^2x}{dt^2}=-9x,$ with initial conditions $x(0)=1$ and $\dfrac{dx}{dt}\bigg \vert_{t=0}=1$ , is$t^2+t+1 \\$$\sin 3t+\df...

A differential equation $\dfrac{di}{dt}-0.2i=0$ is applicable over $−10 < t < 10$. If $i(4) = 10$, then $i(−5)$ is _________.

A solution of the ordinary differential equation $\dfrac{d^{2}y}{dt^{2}}+5\dfrac{dy}{dt}+6y=0$ is such that $y(0) = 2$ and $y(1)= -\dfrac{1-3e}{e^{3}}$. The value of $\df...

Let $y(x)$ be the solution of the differential equation $\frac{d^{2}y}{dx^{2}}-4\frac{dy}{dx}+4y=0$ with initial conditions $y(0)=0$ and $\frac{dy}{dx}\mid _{x=0}=1$ Then...

The solution of the differential equation, for $t 0, y"(t)+2y'(t)+y(t)=0$ with initial conditions $y(0)=0$ and $y'(0)=1$, is ($u(t)$ denotes the unit step function),$te^...

Consider a causal $LTI$ system characterized by differential equation $\frac{dy(t)}{dt}+\frac{1}{6}y(t)=3x(t)$ The response of the system to the input $x(t)=3e^{-\frac{t}...

A function $y(t)$, such that $y(0)=1$ and $y(1)=3e^{-1}$, is a solution of the differential equation$\dfrac{d^{2}y}{dt^{2}}+2\dfrac{dy}{dt}+y=0$. Then $y(2)$ is$5e^{-1}$$...