TI-89HOW TO
Solving Differential Equations with deSolve on the TI-89
Searching for TI-89 differential equations usually starts with one question: can the calculator actually solve them, or only graph them? The TI-89 Titanium solves them symbolically with one command, deSolve(, and this article walks you through it from the first keystroke to a particular solution.
Can the TI-89 solve differential equations?
Yes. deSolve( handles first and second order ordinary differential equations and returns an exact formula, not a table of numbers. Separable, linear and many homogeneous equations come back in one step. What it does not do is guess: you tell it the equation, the independent variable and the dependent variable, and it either finds a closed form or reports that it cannot. For equations without a formula you switch to a numeric method, covered further down.
The full syntax is short:
deSolve(equation, independent variable, dependent variable)
For a textbook problem in y and x, that is always deSolve(…, x, y). Mixing up the order is the most common reason for an error.
Step 1: open deSolve( and type the equation
On the TI-89 Titanium, deSolve( is the last item of the F3 Calc menu on the home screen. You can also open CATALOG and jump to the letter D.
Derivatives are written with the prime symbol, the 2nd function of the = key. One prime is y′, two primes are y′′. Type plain y, not y(x); the third argument already tells the CAS that y depends on x.
Step 2: read the general solution
Close the parentheses after x,y and press ENTER. Without a condition, the answer contains an arbitrary constant written as @1. It plays the role of C in your textbook.
| Input | Result |
|---|---|
| deSolve(y'=x,x,y) | y=x2/2+@1 |
This is the simplest case, y′ = x, where deSolve( does nothing more than integrate. The same pattern holds for harder equations: one @ constant per order, so a second order equation shows @1 and @2.
Step 3: add an initial condition with and
Most homework asks for a particular solution. Put the condition inside the first argument and join it to the equation with the word and. Type and with the alpha letters, with a space before and after it, or pick it from CATALOG. Write the condition in function form, y(0)=1, even though the equation uses plain y.
| Input | Result |
|---|---|
| deSolve(y'+2·y=6 and y(0)=1,x,y) | y=3−2/e2·x |
| deSolve(y'=x·y and y(0)=1,x,y) | y=ex2/2 |
The first row is a linear equation: the solution starts at 1 and settles at 3. The CAS writes the decaying part as a fraction with e2·x in the denominator, which is how the TI-89 Titanium prefers to show negative exponents. The second row is separable, and the separation and both integrals happen inside the one command.
To evaluate a solution at a point, copy the right side and add the with operator, for example |x=1, then press ◆ ENTER for a decimal.
Second order equations
A second order equation needs two conditions to fix both constants, usually a value and a slope at the same point. Chain them with and:
deSolve(y''+y=0 and y(0)=1 and y'(0)=0,x,y)
Type y′′ as y followed by 2nd = twice. Boundary conditions at two different points, such as y(0) and y(1), use exactly the same form. If you leave out the conditions, you get the general solution with @1 and @2, which is useful when the task is to find the form of the solution first.
When there is no formula: euler( and rk23(
Some equations, especially the nonlinear ones from population models and physics, have no closed form that deSolve( can find. On the TI-89 Titanium you then switch the Graph setting in MODE to DIFF EQUATIONS and pick Euler or Runge Kutta in the graph format. Calc89 also offers both methods as commands on the home line: euler( and rk23(. Each takes the right side of y′, the two variables, the interval as a list, the start value and the step size, and returns a matrix whose first row holds the t values and second row the y values.
| Input | Result |
|---|---|
| euler(0.001·y·(100−y),t,y,{0,2},10,1) | 0.01.02.010.010.911.87119 |
| rk23(0.001·y·(100−y),t,y,{0,1},10,1) | 0.01.010.010.9366808323 |
Both rows start at y = 10 on the logistic equation y′ = 0.001·y·(100 − y). After one step, Euler gives 10.9 and Runge Kutta 10.9366808323. Runge Kutta is the more accurate of the two at the same step size; with Euler, smaller steps get you closer at the cost of more rows.
Common mistakes
- Arguments swapped. The independent variable comes first:
x, y. - Stored variables. If y or x holds a number, deSolve( cannot treat y as a function. Run NewProb from the Clean Up menu (2nd F1) before you start.
- Apostrophe instead of prime. On the handheld, use 2nd = for ′.
- Condition as y=1. Initial values need the form y(0)=1.
- Too few conditions. A second order equation with one condition still shows a constant in the answer.
In Calc89
Calc89 is a CAS calculator app intended for use with TI-89 Titanium, and deSolve( takes the same syntax: the prime from 2nd =, conditions joined with and, then x and y. The results are exact, so the rows above look the same on your phone. Next to the solution, Calc89 draws the slope field, which is a quick check that your answer follows the arrows. If the equation is printed in your homework, point the camera scanner at it instead of typing primes and parentheses. The keystroke tutor then shows the key sequence step by step, so you can repeat it on the TI-89 Titanium in class. deSolve(, euler( and rk23( are part of the Pro features. Calc89 is free to download, Pro features as an in-app purchase.
For more worked cases, including slope fields on the handheld, see the guide Differential equations on the TI-89 Titanium.
Frequently asked questions
Can the TI-89 Titanium solve differential equations?
Yes. deSolve( solves first and second order ordinary differential equations symbolically, with or without initial conditions. Equations without a closed form are handled numerically in the Differential Equations graph mode.
What does @1 mean in a deSolve( answer?
It is an arbitrary constant, the C of the general solution. A second constant appears as @2. Add an initial condition with and to replace it by a number.
Why does deSolve( return an error or the input unchanged?
Usually y or x holds a stored value, the arguments are in the wrong order, or an apostrophe was typed instead of the prime symbol. Clear the variables with NewProb and check the order x, y.
Related pages
- /commands/desolve/
- /commands/euler/
- /commands/rk23/
- /guides/ti-89-differential-equations/
- /guides/ti-89-ap-calculus/
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