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(2*t*): *y*”(*t*) + *y*(*t*) = sin(2*t*), *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:

Learn about a named ODE:

Solve a higher-order differential equation:

Solve a vector ODE:

Solve an ODE with differential notation:

Find the solution to a general class of differential equations:

Solve a PDE:

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

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!