Recent questions tagged differential-equations

The equation $\dfrac{d^{2} y}{d x^{2}}-y=0$ has a solution of the form $y=e^{A x}$. The value(s) of $A$ satisfying this is/are:$0$$1$$-1$$-\infty$

A heat exchanger during operation in a bioprocess has a steady temperature of $90^{\circ} \mathrm{C}$. After completion of its operation, it was shut down and it was obse...

​​​​​The output $y(t)$ of a first-order process is governed by the following differential equation\[\tau_{p} \frac{d y}{d t}+y=K_{p} f(t)\]where $\tau_{p}$ is a non-zero ...

​​Let $m$ and $n$ be fixed real numbers. If the function $y(t)=C_{1} e^{t}+C_{2} e^{-t}$ is a solution of$$ \frac{d^{2} y}{d t^{2}}+m \frac{d y}{d t}+n y=0$$for any const...

The solution of the differential equation $\frac{d y}{d x}=y+e^{-x}$ that satisfies $y(0)=-\frac{1}{2}$ is $\_\_\_\_\_\_\_$.$-\frac{1}{2} e^{-\frac{x}{2}}$$-\frac{1}{2} e...

Let $y(x)=x^2 \ln x$ for $x>0$, be a solution of $x^2 \frac{d^2 y}{d x^2}+4 y=\alpha x \frac{d y}{d x}$. Then the value of $\alpha$ is $\_\_\_\_\_\_\_\_\_$.

What is the order of the differential equation given below?$\dfrac{d^{2}y}{dx^{2}} – 6x = 3x^{4} – 2x^{3} + 2$$1$$2$$3$$4$

Consider the ordinary differential equation $\dfrac{dy}{dx} = f(x,y) = 2x^{2} – y^{2}.$ If $y(1)=1,$ the value(s) of $y(1.5),$ using the Euler’s implicit method $\left[y_...

What is the solution of the differential equation $\dfrac{\mathrm{dy} }{\mathrm{d} x}=\dfrac{x}{y}$, with the initial condition, at $x=0, y=1?$$x^{2}=y^{2}+1$$y^{2}=x^{2}...

The Laplace transform of the function $f\left ( t \right )=t^{2}+2t+1$ is $\dfrac{1}{s^{3}}+\dfrac{3}{s^{2}}+\dfrac{1}{s} \\$$\dfrac{4}{s^{3}}+\dfrac{4}{s^{2}}+\dfrac{1}{...

A variable $Y$ is a function of $t$. Given that $Y\left ( t=0 \right )=1$ and $Y\left ( t=1 \right )=2,\dfrac{dY}{dt}$ in the interval $t=\left [ 0,1 \right ]$ can be ap...

Given that $Z=X^{2}+Y^{2}$, the value of $\dfrac{\partial Z}{\partial X}$ for $X=1$ and $Y=0$ is ________________ (answer is an integer).

Which one of the following equations represents a one-dimensional wave equation?$\dfrac{\partial u}{\partial t}=C^{2}\dfrac{\partial ^{2}u}{\partial x^{2}} \\$$\dfrac{\pa...

The Laplace transform $F(s)$ of the function $f(t) = \cos (at)$, where $a$ is constant, _________$\dfrac{s^2}{s^2+a^2} \\$$\dfrac{a}{s^2+a^2} \\$$\dfrac{s}{s^2+a^2} \\$$\...

$\dfrac{d^2y}{dx^2}-y=0$. The conditions for this second order homogeneous differential equation are $y(0)=1$ and $\dfrac{dy}{dx}=3$ at $x=0$. The value of $y$ when $x = ...

For $y=f(x)$, if $\dfrac{d^2y}{dx^2}=0$, $\dfrac{dy}{dx}=0$ at $x=0$, and $y=1$ at $x=1$, the value of $y$ at $x=2$ is __________

Growth of a microbe in a test tube is modeled as $\dfrac{dX}{dt} = rX \bigg (1 – \dfrac{X}{K}\bigg )$, where, $X$ is the biomass, $r$ is the growth rate, and $K$ is the c...

The concentration profile of a chemical at a location $x$ and time $t$, denoted by $c(x,t)$, changes as per the following equation,$$c(x,t)=\frac{c_0}{\sqrt{2\pi Dt}}\exp...

The solution to $\frac {dy}{dx}+y \cot x=\csc x$ is$y=(c+x)\cot x$$y=(c+x)\csc x$$y=(c+x)\csc x\cot x$$y=(c+x)\frac{\csc x}{\cot x}$

The Laplace transform of $f(t) = 2t + 6$ is$\frac{1}{s}+\frac{2}{s^2}$$\frac{3}{s}-\frac{6}{s^2}$$\frac{6}{s}+\frac{2}{s^2}$$-\frac{6}{s}+\frac{2}{s^2}$