Differential equations on the TI-89 Titanium: deSolve( step by step

The three arguments of deSolve(

The syntax is deSolve(equation, independent variable, dependent variable). For an equation in y and x that means deSolve(…, x, y), always in that order. The command is the last item of the F3 Calc menu; you can also find it in CATALOG under D. Command page: deSolve(.

Derivatives of the dependent variable are written with the prime symbol, which is a 2nd function of the = key. Press it once for y′ and twice for y′′. Write y, not y(x): deSolve( knows from the third argument that y depends on x.

First order: the general solution

Without initial conditions, deSolve( returns the general solution with an arbitrary constant written as @1. A second constant in the same session is @2, and so on. The simplest case is a direct antiderivative:

Read the result as y = x²/2 + C. The constant stays symbolic until you add a condition, which is the next step.

Initial conditions with and

To get a particular solution, join the equation and the condition with the word and inside the first argument. Type and with the alpha letters, with a space on each side (alpha (−) is the space), or pick it from CATALOG. The condition is written in function form, y(0)=1, even though the equation itself uses plain y.

The first row is a linear equation with a constant input: the solution starts at 1 and approaches the equilibrium 3. The second is separable, and the CAS does the separation and the integration in one step. Both results are exact. Press ◆ ENTER after |x=1 if you need a decimal value at a point.

Second order equations

Second order equations use y′′, typed as y followed by 2nd = twice. Without conditions the general solution carries two constants, @1 and @2. To fix both, give two conditions joined with and, usually a value and a slope:

deSolve(y''+y=0 and y(0)=1 and y'(0)=0,x,y)

The pattern is the same as for first order: equation first, then each condition, all joined with and, then x and y. Boundary conditions at two different points, such as y(0) and y(1), are written the same way. When an equation has no closed form solution, a numeric method is the next step.

Numeric solutions: euler( and rk23(

Many equations from physics and biology have no closed form. On the TI-89 Titanium, numeric solutions live in the Differential Equations graph mode, where the graph format lets you choose between the Euler and the Runge Kutta method. Calc89 also offers both methods as commands on the home line: euler( and rk23(. Each takes the right side of y′, the variables, the interval as a list, the start value and the step, and returns a matrix: the first row holds the t values, the second the y values.

Both rows solve the same logistic equation from y(0) = 10. Euler with step 1 gives 10.9 after one step; the Runge Kutta method gives 10.9366808323. At the same step size Runge Kutta is the more accurate method; smaller steps bring Euler closer, at the cost of more rows.

Slope fields

On the handheld, slope fields belong to the Differential Equations graph mode: switch Graph to DIFF EQUATIONS in MODE, enter y1′ in the Y= editor and graph. Calc89 draws slope fields too, next to the solution curve, so you can check that a deSolve( result follows the field.

Common mistakes

• Wrong argument order. The independent variable comes before the dependent one: x, y, not y, x.
• A stored y or x. If y holds a number, deSolve( cannot treat it as a function. Run NewProb from Clean Up (2nd F1) first.
• Apostrophe instead of prime. On the handheld, use 2nd = for the prime symbol.
• Condition written as y=1. Initial values need the function form, y(0)=1.

FAQ

Calc89 is an independent app intended for use with TI-89 Titanium. It is not affiliated with or endorsed by Texas Instruments. TI-89 Titanium is a trademark of Texas Instruments.