Ordinary Differential Equations (ODEs) Made Easy

As the winter term kicks into gear, you might start hoping you had an ODE-solving pet monkey as the math and physics problem sets start piling up. Now, we do not offer ODE-solving primates at the moment, but we can help you with your differential equations problem sets. Wolfram|Alpha can solve a plethora of ODEs, each using multiple methods.

Let’s look at the simple ODE y‘(x) = y(x). We see Wolfram|Alpha classifies the ODE, solves it, and provides a family of plots. Notice how four methods are provided with the Step-by-step solution.

How about we model the position of a spring with resting initial position and velocity, and forcing function sin(2t): y”(t) + y(t) = sin(2t), y(0) = 0, y‘(0) = 0. Three methods are provided here for solving this ODE.

An example of an ODE that models the angle of a pendulum over time is y“(t) + sin(y(t)) = 0. The solution for this ODE is in terms of special functions, which is not a problem for Wolfram|Alpha. For small angles, we can approximate this ODE by y”(t) + y(t) = 0.

The coverage of differential equations Wolfram|Alpha provides goes much deeper.

Given a function, we can calculate an ODE such that the given function is a solution:

• differential equation sqrt(x) sin(x)

Learn about a named ODE:

• Laguerre differential equation

Solve a higher-order differential equation:

• 5 y”” + 4 y”” + 3 y”’ + 2 y” + 1 y‘ + 0 y = 0

Solve a vector ODE:

• x‘ = {{4, -2}, {1, 6}}.x

Solve an ODE with differential notation:

• arctan(x) dx = arccot(y) dy

Find the solution to a general class of differential equations:

• y‘(x) + p(x) y(x) = q(x)

Solve a PDE:

• d^2/dx^2 f(x, y) + d^2/dy^2 f(x, y) = 0

Calculate solutions to an ODE that are implicit or in terms of special functions:

• y” + 2y‘y + y = 0
• (x^2 – x) y”(x) + x y‘(x) + 1 = 0

Interactivity is available for Wolfram|Alpha Pro users. Check out this complex-plane solution curve for free:

Wolfram|Alpha can do more with differential equations, such as wronskian of cos(t)+1, sin(t), laplace transform of t^2*sin(t), 1D harmonic oscillator Schrödinger equation, free particle in 2D, throwing with quadratic drag, or forward euler method y‘(x) + y(x) = x, y(0) = 1, h = 0.01.

So now you don’t have to choose between violating the dorm’s pet policy and getting your problem set done in a timely fashion. Use Wolfram|Alpha to not only solve your differential equations, but to receive help understanding each step of the solution so that you are better prepared for exams!