Module 18  Antiderivatives as Indefinite Integrals and Differential Equations  
Introduction  Lesson 1  Lesson 2  Lesson 3  Lesson 4  SelfTest  
Lesson 18.3: Solving Differential Equations Analytically  
This lesson investigates finding analytical solutions to differential equations by using the TI89's deSolve command and explores finding solutions to initialvalue problems. An equation containing a derivative such as y' = 2x, where y' denotes the derivative , is called a differential equation. Such equations occur often in realworld applications. Finding Solutions to Differential Equations A solution to a differential equation is a function that satisfies the differential equation and there may be more than one solution. 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 described by the function y = x^{2} + C where C represents any real number. This family of functions is called the general solution to the differential equation. Using the deSolve Command The TI89 deSolve command can be used to find general solutions to simple differential equations like y' = 2x and to more complicated differential equations as well. Solve y' = 2x by following the procedure below.
Note that is above .
The solution may be verified by substituting the result into the original differential equation.
18.3.1 Use the deSolve command to find the general solution to the differential equation y' = e^{x} sin x Click here for the answer.
18.3.2 Use deSolve to find the general solution to the differential equation y' = y Click here for the answer. Solving InitialValue Problems Solve 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. Solve the initialvalue problem y' = 2x and y(0) = 1.
Recall that "and " is found in the catalog. The solution to an initialvalue problem is called a particular solution to the differential equation. The initial condition selects a particular solution from the family of general solutions.
18.3.3 Solve the initialvalue problem y' = 2x and y(2) = 1.
18.3.4 Solve the initialvalue problem and y(1) = 3. Don't forget to insert a multiplication sign between y and ln(x). Click here for the answer. 

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