Graphing a Second Order Differential Equation

This section explains how to input a second order differential equation, draw a phase plane, and graph the solution curve(s). With this application, a second order differential equation is input in the form of a set of two first order differential equations.

•A phase plane is the family of solutions of either a second order differential equation or two first order differential equations of the form x’ = dx/dt = f (x, y) and y’ = dy/dt = g(x, y). A single second order differential equation can also be graphed, but it must be written as two first order differential equations.

•You can overlay, onto the phase plane, solution curves of the second order differential equation input on the [DiffEq] tab for given initial conditions.

uTo input a second order differential equation and draw a phase plane

0503 To input {x’ = x, y’ = −y} and draw its phase plane

uTo input initial conditions and graph the solution curves

0504 After performing the operation under example 0503 , to graph the solution curve of the initial condition (xi, yi) = (1, 1)

Independent variable minimum value (tmin) = −7.7, maximum value (tmax) = 7.7, and initial value (t0) = 0

Graphing an Nth-order Differential Equation

This section explains how to graph the solution curve(s) for an nth-order (higher order) differential equation based on specified initial conditions. With this application, an nth-order differential equation is input in the form of a set of multiple first order differential equations.

Note: For nth-order differential equations, only solution curves are drawn.

uTo input an nth-order differential equation and initial conditions, and then graph the solution curves

0505 To specify the three initial conditions (xi, y1i, y2i) = (0, −1, 0), (0, 0, 0), (0, 1, 0) for the differential equation y” = x − y, and graph its solution curves

Chapter 5: Differential Equation Graph Application

120