## Differential Equations Calculator Applet

This is a general purpose tool to help you solve differential equations numerically by any one of several methods.

## Using the Applet

1. Be patient. The applet may take a long time to load and start.
2. Click inside the text area under Equations.
3. Type into it the differential equation you want to solve, for example

y' = x y^2 - cos(y)
4. Press the Enter key.

Points to keep in mind:

• The derivative is denoted by an apostrophe '.
• The independent variable is normally x , but you can change this by clicking on the choice box after Independent Variable.
• You can use either * or space for multiplication. For example, 4 x y and 4*x*y are both OK. But don't just write xy without a space.
• Use the caret ^ (on the 6 key) for exponents.
• Use exp(x) instead of e^x.
• Functions such as sin and exp need parentheses around their arguments.
• There is no distinction between upper and lower case.
5. After you enter a differential equation, an input area for the dependent variable (here y ) will appear under Initial Values. You can enter into it the initial value y0 of that variable.
6. The initial and final values of the independent variable are set in the Initial X and Final X boxes on the left side. Thus if your initial condition was at x0 = 1 and you wanted the solution at xf = 3 you would enter 1 for Initial X and 3 for Final X.
7. Click the left mouse button on Euler, and a list of numerical methods drops down. Click on the one you want to use.
8. At Number of steps enter the number N of steps of the method to perform. Thus the step size will be dx = (xf - x0)/N .
9. When everything is set up, click the Go button. This will find an approximation to the solution of the initial value problem, and show you the final values of the variables under Final Values.
10. To see the table of saved values, click the Results button. During the solution process, values of the variables are recorded at Number of saved values values of the independent variable (in addition to the initial values). This number can never be more than Number of steps. If it divides Number of steps evenly, the values are recorded at equal intervals.
11. It is possible to enter more than one equation, and they don't all have to be differential equations. For example, if you want to compare the values of y to some known function of x, say x2, you could add a new equation (by clicking the Insert equation button), and enter the equation

z = y - x^2

Then when you click Go and Results, the applet will compute and show the values of z together with those of x and y. Up to three columns of data can be shown in the Results window, representing any three variables. Click on the names of the variables there to choose which variables to show, and in what order. Generally, not all the data are visible at one time: use the scroll bars to see the rest.

On some computers, the applet window may not be wide enough to accommodate the applet. If you see "Try this at a larger size" in the bottom left of the applet window, try the wider version of this page.