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}{\partial y^2}\frac{\partia...

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 decimal place...

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

Consider the second-order linear ordinary differential equation $\rm x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}-y=0, x\ge1$ with the initial conditions $\rm y(x=1)=6, \left.\frac{dy}{dx}\...

If y(x) satisfies the differential equation (sin x) dy/dx + y cos x = 1, subject to the condition y(π/2) = π/2, then y(π/6) is

Consider the following differential equation $(1 + y) \frac{dy}{dx} = y$. The solution of the equation that satisfies the condition $y(1) = 1$ is

A differential equation is given as $x^2 \frac{d^2 y}{dx^2} - 2x \frac{dy}{dx} + 2y = 4$. The solution of the differential equation in terms of arbitrary constants $C_1$ and $C_2$...

Consider the following partial differential equation for u(x, y) with the constant c > 1: \frac{\partial u}{\partial y} + c \frac{\partial u}{\partial x} = 0 Solution of this equat...

The differential equation \frac{d^2y}{dx^2} + 16y = 0 for y(x) with the two boundary conditions \frac{dy}{dx}|_{x=0} = 1 and \frac{dy}{dx}|_{x=\frac{\pi}{2}} = -1 has

Consider the differential equation $3y''(x)+27y(x)=0$ with initial conditions $y(0)=0$ and $y'(0)=2000$. The value of y at x = 1 is _________.

The differential equation $$\,{{{d^2}y} \over {d{x^2}}} + 16y = 0$$ for $$y(x)$$ with the two boundary conditions $${\left. {{{dy} \over {dx}}} \right|_{x = 0}} = 1$$ and $${\left....

Find the solution of $${{{d^2}y} \over {d{x^2}}} = y$$ which passes through origin and the point $$\left( {ln2,{3 \over 4}} \right)$$

The matrix form of the linear system $${{dx} \over {dt}} = 3x - 5y$$ and $$\,{{dy} \over {dt}} = 4x + 8y\,\,$$ is

Consider the differential equation $${{dy} \over {dx}} = \left( {1 + {y^2}} \right)x\,\,.$$ The general solution with constant $$'C'$$ is

Given that $$\mathop x\limits^{ \bullet \bullet } + 3x = 0$$ and $$x\left( 0 \right) = 1,\,\,\mathop x\limits^ \bullet \left( 0 \right) = 1,$$ What is $$x(1)$$ ________.

The partial differential equation $$\,\,{{{\partial ^2}\phi } \over {\partial {x^2}}} + {{{\partial ^2}\phi } \over {\partial {y^2}}} + {{\partial \phi } \over {\partial x}} + {{\p...

For $$\,\,\,{{{d^2}y} \over {d{x^2}}} + 4{{dy} \over {dx}} + 3y = 3{e^{2x}},\,\,$$ the particular integral is

If $${x^2}\left( {{{d\,y} \over {d\,x}}} \right) + 2xy = {{2\ln x} \over x}$$ and $$y(1)=0$$ then what is $$y(e)$$?

The equation $$\,\,\,{{{d^2}u} \over {d{x^2}}} + \left( {{x^2} + 4x} \right){{dy} \over {dx}} + y = {x^8} - 8\,\,{u \over {{x^2}}} = 8.\,\,\,$$ is a

The particular solution for the differential equation $${{{d^2}y} \over {d{t^2}}} + 3{{dy} \over {dx}} + 2y = 5\cos x$$ is

A differential equation of the form $${{dy} \over {dx}} = f\left( {x,y} \right)\,\,$$ is homogeneous if the function $$f(x,y)$$ depends only on the ratio $${y \over x}$$ (or) $${x...

The differential $$\,\,\,{{{d^2}y} \over {d{x^2}}} + {{dy} \over {dx}} + \sin y = 0\,\,$$ is

The differential equation $${y^{11}} + y = 0\,$$ is subjected to the conditions $$y(0) = 0,$$ $$\,\,\,y\left( \lambda \right) = 0.\,\,$$ In order that the equation has non-trivial...

Given the differential equation $${y^1} = x - y$$ with initial condition $$y(0)=0.$$ The value of $$y(0.1)$$ calculated numerically upto the third place of decimal by the $${2^{nd}...