differential equations

A partial differential equation is given below. $\frac{\partial^2 u}{\partial x^2} - \frac{\partial^2 u}{\partial y^2} = 0$ Possible solution(s) is/are:

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

The "order" of the following ordinary differential equation is ______. $d^3y/dx^3 + (d^2y/dx^2)^6 + (dy/dx)^4 + y = 0$

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 decima...

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 off to 1 decimal p...

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

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

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 conc...

In the differential equation $\frac{dy}{dx} + \alpha xy = 0$, $\alpha$ is a positive constant. If $y = 1.0$ at $x = 0.0$, and $y = 0.8$ at $x = 1.0$, the value of $\alpha$ is _____...

The solution of the differential equation $\frac{d^3y}{dx^3} - 5.5 \frac{d^2y}{dx^2} + 9.5 \frac{dy}{dx} - 5 y = 0$ is expressed as $y = C_1e^{2.5 x} + C_2e^{\alpha x} + C_3e^{\bet...

The solution of the second-order differential equation $\frac{d^2y}{dx^2} + 2\frac{dy}{dx} + y = 0$ with boundary conditions $y(0) = 1$ and $y(1) = 3$ is

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

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

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

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

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

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...

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=2...

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

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?

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

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

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

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

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

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