Module 20  Antiderivatives as Indefinite Integrals and Differential Equations  
Introduction  Lesson 1  Lesson 2  Lesson 3  Lesson 4  SelfTest  
Lesson 20.2: Solving Differential Equations Analytically and Graphically  
This lesson investigates finding solutions to differential equations by analytical and graphical methods. It also explores finding solutions to initialvalue problems. You will use a TI83 program called EULERG to graph solutions to differential equations. Definition of a Differential Equation An equation containing a derivative is called a differential equation. For example, y ' = 2x is a differential equation because it contains the derivative y '. Such equations occur often in realworld applications. Solutions to Differential Equations A solution to a differential equation is a function that satisfies the differential equation. There is usually more than one solution to any differential equation. Each of the functions shown below is a solution to the differential equation y ' = 2x because each makes the differential equation true. y_{1} = x^{2} y_{2} = x^{2} + 1 y_{3} = x^{2}  3 These particular solutions, as well as all other solutions to the differential equation y ' = 2x, can be found by using the power rule for integrals.
where C can be any real number. This family of functions is called the general solution to the differential equation.
The solution may be verified by substituting the solution into the original differential equation. Solutions to y ' = f (x) Solutions to differential equations of the form y ' = f (x) may be found by evaluating but solutions to differential equations of other forms require other techniques. 20.2.1 Solve the differential equation by using the power rule for integrals. Click here for the answer. Initial Conditions and InitialValue Problems Suppose you need the particular function that satisfies the differential equation y ' = 2x with the condition that y(0) = 1, that is, y = 1 when x = 0. The condition y(0) = 1 is called an initial condition and a differential equation with an initial condition is called an initialvalue problem. The general solution to the differential equation y ' = 2x is y = x^{2} + C. The value of C can be found by substituting the initial values x = 0 and y = 1.
1 = 0^{2} + C So, the desired particular solution is y = x^{2} + 1. Any one specific solution of a differential equation is called a particular solution because it is just one solution out of many. In elementary initial value problems, the initial condition selects one particular solution from the family of all solutions. 20.2.2 Solve the initialvalue problem, , y(1) = 1. Click here for the answer. Graphical Solutions to Differential Equations You can use a TI83 program called EULERG to graph approximate solutions to initialvalue problems without finding the analytic solution. 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.
Graphing Solutions to Differential Equations Graph an approximate solution to the initial value problem y' = 2x with the condition that y(0) = 1 by following the procedure outlined below. The program EULERG requires that you enter the differential equation into Y_{1} in the Y= editor and select an appropriate viewing window for the solution before you run the program.
Running EULERG
After you enter the STEP SIZE the TI83 will display a graph. This is the graph of y = x^{2} + 1, the solution to the initial value problem.
20.2.3 Use EULERG to graph the solution to the initial value problem
with y(1) = 1 in a Click here for the answer. y' as a Function of y EULERG can also be used to graph solutions to differential equations that depend on both x and y. Graph the solution to the initialvalue problem y' = 2y with y(0) = 1 using the EULERG program.
This is a graphical solution to the initial value problem. The analytical solution is y = e^{2x}. Graph y = e^{2x} to compare with the solution given by EULERG.
Note that the analytical solution to y' = 2y was not found by evaluating an indefinite integral. Several types of differential equations require other techniques to produce solutions and many differential equations do not have analytically determined solutions at all.


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