Recent questions tagged ordinary-differential-equation

Periodic function $f(x)$ is given below.\[f(x)=\left\{\begin{array}{lll}-1, & \text{when} & -\pi < x < 0 \\1, & \text{when} & 0 < x < \pi\end{array}\right\};\; f(x+2\pi)...

Consider differential equation $\dfrac{d y}{d x}+x y=x$ with the condition as $y=0$ at $x=0$. The value of $y$ at $x=1.0$ is $\_\_\_\_$ (rounded off to two decimal places...

An ordinary differential equation is given below.\[x^{2} \frac{d^{2} y}{d x^{2}}=6 y\]Considering $a$ and $b$ as arbitrary constants, the general solution of the equation...

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

Consider the differential equation given below. Using the Euler method with the step size $(h)$ of $0.5$, the value of $y$ at $x=1.0$ is equal to $\_\_\_\_\_$ (rounded of...

A $2 \mathrm{~m} \times 2 \mathrm{~m}$ tank of $3 \mathrm{~m}$ height has inflow, outflow and stirring mechanisms. Initially, the tank was half-filled with fresh water. A...

The solution of the differential equation$\frac{d^3 y}{d x^3}-5.5 \frac{d^2 y}{d x^2}+9.5 \frac{d y}{d x}-5 y=0$is expressed as $y=C_1 e^{2.5 x}+C_2 e^{\alpha x}+C_3 e^{\...

The differential equation,$$\frac{\mathrm{d} u}{\mathrm{~d} t}+2 t u^2=1,$$is solved by employing a backward difference scheme within the finite difference framework. The...

For the equation $$\frac{d^{3}y}{dx^{3}} + x \left ( \frac{dy}{dx} \right )^{\frac{3}{2}}+ x^{2}y=0$$ the correct description isan ordinary differential equation of order...

Consider the differential equation $$\frac{\mathrm{dy} }{\mathrm{d} x} = 4 \left ( x + 2 \right ) - y$$ For the initial condition $y = 3$ at $x = 1$, the value of $y$ at ...

If $\text{k}$ is a constant, the general solution of $\dfrac{dy}{dx}-\dfrac{y}{x}=1$ will be in the form of$y=x\text{ ln}(kx)$$y=k\text{ ln}(kx)$$y=x\text{ ln}(x)$$y=xk\t...

The solution of the second-order differential equation $\dfrac{d^{2}y}{dx^{2}}+2\dfrac{dy}{dx}+y=0$ with boundary conditions $y\left ( 0 \right )=1$ and $y\left ( 1 \righ...

For the Ordinary Differential Equation ${\large\frac{d^2x}{dt^2}}-5{\large\frac{dx}{dt}}+6x=0$, with initial conditions $x(0)=0$ and ${\large\frac{dx}{dt}}(0)=10$, the s...

The ordinary differential equation $\dfrac{d^2u}{dx^2}$$- 2x^2u +\sin x = 0$ islinear and homogeneouslinear and nonhomogeneousnonlinear and homogeneousnonlinear and nonho...

An ordinary differential equation is given below$$6\dfrac{d^2y}{dx^2}+\frac{dy}{dx}-y=0$$The general solution of the above equation (with constants $C_1$ and $C_2$), is$y...

Consider the following second-order differential equation: $y’’ – 4y’+3y =2t -3t^2$. The particular solution of the differential solution equation is$ – 2 -2t-t^2$$ – 2t-...

Consider the ordinary differential equation $x^2 \dfrac{d^2y}{dx^2} – 2x \dfrac{dy}{dx} +2y=0$. Given the values of $y(1)=0$ and $y(2)=2$, the value of $y(3)$ (round off ...

The Laplace transform of $\sin h (\text{at})$ is $\dfrac{a}{s^2-a^2} \\$$\dfrac{a}{s^2 + a^2} \\$$\dfrac{s}{s^2-a^2} \\$$\dfrac{s}{s^2+a^2}$

An ordinary differential equation is given below;$\left ( \dfrac{dy}{dx} \right ) (x \text{ ln } x)=y$The solution for the above equation is(Note: $K$ denotes a constant ...

The respective expressions for complementary function and particular integral part of the solution of the differential equation $\dfrac{d^4y}{dx^4}+3 \dfrac{d^2y}{dx^2} =...

The solution of the equation $\dfrac{dQ}{dt} +Q =1$ with $Q=0$ at $t=0$ is$Q(t)=e^{-t}-1$$Q(t)=1+ e^{-t}$$Q(t)=1 -e^t$$Q(t)=1- e^{-t}$

Consider the equation $\dfrac{du}{dt}=3t^2+1$ with $u=0$ at $t=0$. This is numerically solved by using the forward Euler method with a step size, $\Delta t=2$. The absolu...

Consider the following second order linear differential equation$$\dfrac{d^2y}{dx^2} = -12x^2 +24 x – 20$$The boundary conditions are: at $x=0, \: y=5$ and at $x=2, \: y=...

Consider the following differential equation:$$x(y\:dx +x\:dy) \cos \dfrac{y}{x}=y(x\:dy-y\:dx) \sin \dfrac{y}{x}$$Which of the following is the solution of the above equ...

The integrating factor for the differential equation $\dfrac{dP}{dt}+k_2P=k_1L_0e^{-k_1t}$ is$e^{-k_1t} \\$$e^{-k_2t} \\$$e^{k_1t} \\$$e^{k_2t}$

The solution of the ordinary differential equation $\dfrac{dy}{dx}+2y=0$ for the boundary condition, $y=5$ at $x=1$ is$y=e^{-2x}$$y=2e^{-2x}$$y=10.95 e^{-2x}$$y=36.95 e^{...

The Laplace transform $F(s)$ of the exponential function, $f(t) = e^{at}$ when $t \geq 0$, where $a$ is a constant and $(s-a) >0$, is$\dfrac{1}{s+a} \\$$\dfrac{1}{s-a} \\...

The solution of the equation $x \frac{dy}{dx} +y = 0$ passing through the point $(1,1)$ is$x$$x^2$$x^{-1}$$x^{-2}$