(a)
(b)
was referred to in the movie Stand and Deliver.
Differential Equations: Motion Problems
An equation involving a derivative is called a differential equation. In Examples
48 and 49, we solved two simple differential equations. In each one we were
given the derivative of a function and the value of the function at a particular
point. The problem of finding the function is called an initial-value problem, and
the given condition is called the initial condition.
In Examples 50 and 51, we use the velocity (or the acceleration) of a particle
moving on a line to find the position of the particle. Note especially how the
initial conditions are used to evaluate constants of integration.
Example 50 __
The velocity of a particle moving along a line is given by v(t) = 4t^3 − 3t^2 at time
t. If the particle is initially at x = 3 on the line, find its position when t = 2.
SOLUTION: Since
Since x(0) = 0^4 − 0^3 + C = 3, we see that C = 3, and that the position function is
x(t) = t^4 − t^3 + 3. When t = 2, we see that
x(2) = 2^4 − 2^3 + 3 = 16 − 8 + 3 = 11.Example 51 __
Suppose that a(t), the acceleration of a particle at time t, is given by a(t) = 4t − 3,
that v(1) = 6, and that f (2) = 5, where f (t) is the position function.
Find v(t) and f (t).
Find the position of the particle when t = 1.
SOLUTIONS: