Suppose a rock falls from rest from a height of
meters and the only force acting on it is gravity. Find an equation for the velocity
as a function of time, measured in meters per second.
A natural question to ask after solving this type of problem is how high the object will be above Earth’s surface at a given point in time. Let
denote the height above Earth’s surface of the object, measured in meters. Because velocity is the derivative of position (in this case height), this assumption gives the equation
An initial value is necessary; in this case the initial height of the object works well. Let the initial height be given by the equation
Together these assumptions give the initial-value problem
If the velocity function is known, then it is possible to solve for the position function as well.
Height of a moving baseball
A baseball is thrown upward from a height of
meters above Earth’s surface with an initial velocity of
and the only force acting on it is gravity. The ball has a mass of
kilogram at Earth’s surface.
Find the position
of the baseball at time
What is its height after
seconds?
We already know the velocity function for this problem is
The initial height of the baseball is
meters, so
Therefore the initial-value problem for this example is
To solve the initial-value problem, we first find the antiderivatives:
Next we substitute
and solve for
Therefore the position function is
The height of the baseball after
is given by
Therefore the baseball is
meters above Earth’s surface after
seconds. It is worth noting that the mass of the ball cancelled out completely in the process of solving the problem.
A differential equation is an equation involving a function
and one or more of its derivatives. A solution is a function
that satisfies the differential equation when
and its derivatives are substituted into the equation.
The order of a differential equation is the highest order of any derivative of the unknown function that appears in the equation.
A differential equation coupled with an initial value is called an initial-value problem. To solve an initial-value problem, first find the general solution to the differential equation, then determine the value of the constant. Initial-value problems have many applications in science and engineering.
Determine the order of the following differential equations.
