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{\partial\varphi}{\partial x} + \left(\frac{\par...

If $x(t)$ satisfies the differential equation $t\frac{dx}{dt} + (t - x) = 0$ subject to the condition $x(1) = 0$, then the value of $x(2)$ is ________ (rounded off to 2 decimal places).

Consider the second-order linear ordinary differential equation $x^2 \frac{d^2y}{dx^2} + x \frac{dy}{dx} - y = 0, x \geq 1$ with the initial conditions $y(x = 1) = 6, \frac{dy}{dx}|_{x=1} = 2$. The value of y at x = 2 eq...

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

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 $C_2$ is

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 _________.