Tutorial Solving Differential Equations

To solve the differential equation dy/dt = f(t, y) = -0.2y(y-3) subject to the initial condition y(0) = 1,

Select Diff Eq from the Graph menu to use the Differential Equation Solver.
Select New Differential Eqn — First Order from the Work menu.
Type -0.2y(y-3) into the first cell of the Work window. Press the Enter or Return key.
Click in the Graph window to make it active and click the Axes button.
Click in the Domain & Range window to make it active and type 0 into the t-cell and 1 into the y-cell. Click the Enter button or press the Enter key.
You can also click at any point in the Graph window to enter an initial condition and draw the corresponding solution curve. Try this.

To erase the drawn solutions,

To draw the slope field,

To draw a family of solutions,

Click on the Family of Solutions tool on the tool palette in the Graph window. It is the first tool on the right.
Click in the Graph window at the initial t-value that you want. For example, click on the y-axis for an initial t-value of 0.

The harmonic oscillator with damping is modeled by the second-order differential equation

y'' = -ks*y - kd*y'

where

y = the displacement of the free end of the spring from rest,

ks = the spring coefficient, and

kd = the damping coefficient.

To solve this second-order differential equation with

kd = 0.5, ks = 2, y(0) = 3, and y'(0) = 0,

Select New Differential Eqn — Second Order from the Work menu.
Type -2y-0.5y' into the first cell of the Work window. Press the Enter or Return key.
Click in the Graph window to make it active and click the Axes button.
If necessary, click on the Solution Curve tool on the tool palette in the Graph window. It is the second tool from the right.
Click in the Domain & Range window to make it active and type 0 into the t-cell, 3 into the y-cell, and 0 into the y'-cell. Click the Enter button or press the Enter key.
You can also click at any point in the Graph window to enter an initial value and draw the corresponding solution curve (be sure that a value has been entered into the y'-cell). Try this.

To draw the solutions of the harmonic oscillator in the phase plane,