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If f (t) is a continuous function, then the general solution of the ODE
a table of integrals comes is y(t) = F(t) + C, where F(t) is an antiderivative of f. For example, the m handy toe. general solution of dy/ dt = sin t is y =— cos t + C.
Separation of Variables
If you can write a differential equation in the form
¦jt = fW8(y)
then wherever g( y) = 0 you can rewrite it as
g( y) dt
= f (t)
dy = f(t) dt
Finding a Solution Formula
Keep that table of integrals
Figure 2.2: Four solution curves of dy/dt = sin t/(3y2 + 1) through the marked initial points.
If H(y) is an antiderivative of 1/g(y) and F(t) is an antiderivative of f(t), then a solution y( t) of the ODE solves the equation
H( y(t)) = F(t) + C
for some constant C.
Here’s an example of a separable ODE:
dy sin t
dt 3y2 + 1 W
Separating the variables and finding the antiderivatives, we see that
('3j/2+ l^~dt =Sint
y3 + y =— cos t + C (4)
We won’t attempt to express a solution y(t) directly in terms of t (and C),
but we can check that formula (4) is correct by differentiating each side with
respect to t. This gives
3f%+% = sint
which has the form of ODE (3) if we divide each side by 3y2 + 1. Figure 2.2 shows solution curves of ODE (3) through the initial points (0, -1.5), (0, -1), (0, 0), (0, 1). The curves were plotted by using ODE Architect to solve ODE (3) with the given initial data.
Solution formulas are useful, but they exist only for a small number of ODEs of special forms. That’s where numerical solvers like ODE Architect come in—they don’t need solution formulas.
The eight steps are described in Chapter 1.
¦ The Juggler
So the juggler’s ODE for vertical motion is h' = —32.
In the multimedia module h0 = 4.5 ft.
A mathematical model is a system of mathematical equations relating specific variables that represent some aspect of a natural process. Modeling involves several steps:
1. State the problem and its context.
2. Identify and assign variables.
3. State the laws that govern the relationships between the variables.
4. Translate the laws into equations.
5. Solve the resulting equations.
6. Interpret and test the solutions in the context of the natural environment.
7. Refine the model until it predicts the empirical data.
8. Interpret the implications of the model.
The models we consider all involve ODEs.
You can observe the modeling process in the following juggler problem.
1. Find an ODE that describes the height of a ball between the time it leaves the juggler’s hand, moving vertically upward, and the time it falls back into the hand.
2. Let t = time (in seconds), h = height of the ball above the floor (in feet), v = velocity (in ft/sec), and a = acceleration (in ft/sec2).
3. Apply Newton’s second law of motion to the ball: the mass m of a body times its acceleration is equal to the sum of all of the forces acting on the body. We treat the ball as a point mass encountering negligible air resistance (drag) so the only force acting on the ball is that due to gravity, which acts downward.
4. By Newton’s second law, we have that ma = —mg, where g = 32 ft/sec2 is the acceleration due to gravity near the surface of the earth, and the minus sign indicates the downward direction of the gravitational force. Since the ball’s acceleration is a = v' where v is its velocity, and v = h', we can model the ball’s motion by h" = —32. The initial height h0 of the ball is that of the juggler’s hand above the floor when the ball is launched upward, and that is easy to measure. The initial velocity v0 is harder to measure directly; it is simpler to solve the model first and then experiment to deduce a reasonable value for v0.
5-8. Solving and testing are up to you. See Figure 2.3 for graphs of h(t) corresponding to h0 = 4 ft and five values of v0.