Examples for

Differential Equations

A differential equation is an equation involving a function and its derivatives. It can be referred to as an ordinary differential equation (ODE) or a partial differential equation (PDE) depending on whether or not partial derivatives are involved. Wolfram|Alpha can solve many problems under this important branch of mathematics, including solving ODEs, finding an ODE a function satisfies and solving an ODE using a slew of numerical methods.
Ordinary Differential Equations

Solve an ODE or find an ODE a function satisfies.

Solve a linear ordinary differential equation:

y'' + y = 0
w"(x)+w'(x)+w(x)=0

Specify initial values:

y'' + y = 0, y(0)=2, y'(0)=1

Solve an inhomogeneous equation:

y''(t) + y(t) = sin t
x^2 y''' - 2 y' = x

Solve an equation involving a parameter:

y'(t) = a t y(t)

Solve a nonlinear equation:

f'(t) = f(t)^2 + 1
y"(z) + sin(y(z)) = 0

Find differential equations satisfied by a given function:

differential equations sin 2x
differential equations J_2(x)
More examples
More examples
Numerical Differential Equation Solving

Numerically solve a differential equation using a variety of classical methods.

Solve an ODE using a specified numerical method:

Runge-Kutta method, dy/dx = -2xy, y(0) = 2, from 1 to 3, h = .25
{y'(x) = -2 y, y(0)=1} from 0 to 2 by implicit midpoint

Specify an adaptive method:

solve {y'(x) = -2 y, y(0)=1} from 0 to 10 using r k f algorithm
More examples