Module 22  Differential Equations and Euler's Method  
Introduction  Lesson 1  Lesson 2  Lesson 3  SelfTest  
Lesson 22.3: SecondOrder Differential Equations  
In Module 20 you used EULERG and EULERT to solve firstorder differential equations. In this lesson you will learn to use another program called DIFF2G to solve secondorder differential equations. Defining a SecondOrder Differential Equation A secondorder differential equation contains a second derivative. For example, if y is the height above ground of a falling object then the firstorder derivative y ' is the object's velocity and the second derivative y " is its acceleration. The equation y " = k is a secondorder differential equation that represents the movement of an object that has constant acceleration k. Initialvalue problems that involve a secondorder differential equation have two initial conditions. Solving Elementary SecondOrder Differential Equations Suppose that an object is dropped from a height of 48 feet. Because it is dropped without thrust, the initial velocity is zero. The only force acting on the object is gravity, which accelerates it downward 32 feet per second squared. If y is the height of the object above the ground at time t, then y ' is the velocity of the object and y " is its acceleration. The equation for the height of the object can be found by solving two first order differential equations. Because acceleration is the derivative of velocity, the secondorder differential equation for acceleration, y " = a = 32, can be rewritten as a firstorder differential equation.
Because velocity is the derivative of height, the equation for velocity v = –32t can be rewritten as a first order differential equation.
48 = 16 · 0 + C C = 48 The height of the object at time t is given by y = 16t^{2} + 48. Using The Solution The object's height above the ground at t = 1.3 seconds after it was dropped can be found by substituting t = 1.3 in y = 16t^{2} + 48. So the height of the object at t = 1.3 is 20.96 feet. Graphical and Numerical Solution The program DIFF2G can be used to find visual and numerical solutions to secondorder differential equations without finding the analytic solution. The program takes a secondorder differential equation like y " = h(x) and two initial conditions and produces graphs and tablesofvalues for both y ' = g(x) and y = f(x). Downloading the Program to Your Computer
Transferring the Program to the TI83 Click here to get information about how to obtain the needed cable and review the procedure to transfer the program from your computer to your calculator.
Using DIFF2G DIFF2G can be used to solve y " = – 32 with initial conditions y(0) = 48 and y '(0) = 0.
After entering the step size, the program will take a few minutes to calculate the solution. When the calculations are complete you should see two graphs. 22.3.1 One of the graphs is height and the other is velocity. Which is which? Click here for the answer. The graph can be used to approximate the height at t = 1.3.
DIFF2G gives an approximation for y(1.3) of 20.752 feet. This is fairly close to the exact value of 20.96 feet found earlier in this lesson.
22.3.2 Approximate the velocity of the object just before it hits the ground by tracing on the height curve to the xintercept then pressing to jump to the velocity graph. Look at the table generated by DIFF2G to support the values found by tracing. Click here for the answer.


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