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# Check the analytical solution of a differential equation                              with your Ti-84 plus (CE)

## Check the analytical solutions of first-order differential equations, which are useful for checking transients in electrical engineering or dynamics in mechanical engineering.

The method we explain differs slightly from the method used for derivatives and integrals.

Step 1 : Solve the differential equation with the program DV1ETRP.8XP. For information about this program, go to

Step 2 : Edit Y2 or Y1 to fill in the analytical solution you have derived.
Step 3 : The range Xmin to Xmax is already determined when running the differential equation program.
Step 4 : Use the "graph" command to view the graphic solutions
The difference between Ti-84 solution and the analytical solution should be zero or close to zero for the whole range Xmin to Xmax; if so,  the analytical solution is correct. With an example, we will show the method. Suppose you have the following differential equation with Y(0)=0 and an incorrectly derived analytical solution (sorry, I made a mistake).

The analytical solution is filled in the Y editor Y1 or Y2. The next step is to solve the differential equation with the Ti-Basic program DV1ETRP.8xp, and we choose a range Xmin = 0 and Xmax=2.1 (about 8 times the time constant of 0.25). The steps are visualized below. The graph shows a significant difference between my analytical solution and the solution calculated by the program DV1ETRP. Either the program is wrong or my analytical solution is wrong. It appeared that my solution was incorrect.

After trying again I found an error in my solution, so we checked again. The correct solutions is :

The correct solution is filled in Y1

Now the analytical function and the plot1 calculated by DV1ETRP have almost the same value for x. For x=1 the difference is 1.39574-1.39195=0.00379=0.27%. We can conclude that our analytical solution is correct.

For second-order differential equations, the same method can be used. An example of a second order differential equation solution with analytical check is shown on the page:

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