Differential equationA differential equation (d.e.) is any equation involving a derivative. In §E of
Chapter 5 we solved some simple differential equations. In Example 50, we were
given the velocity at time t of a particle moving along the x-axis:
From this we found the antiderivative:
x(t) = t^4 − t^3 + C. (2)If the initial position (at time t = 0) of the particle is x = 3, then
x(0) = 0 − 0 + C = 3,and C = 3. So the solution to the initial-value problem is
x(t) = t^4 − t^3 + 3. (3)
A solution of a d.e. is any function that satisfies it. We see from (2) above that
the d.e. (1) has an infinite number of solutions—one for each real value of C. We
call the family of functions (2) the general solution of the d.e. (1). With the
given initial condition x(0) = 3, we determined C, thus finding the unique
solution (3). This is called the particular solution.
Note that the particular solution must not only satisfy the differential equation
and the initial condition, but the function must also be differentiable on an
interval that contains the initial point. Features such as vertical tangents or
asymptotes restrict the domain of the solution. Therefore, even when they are
defined by the same algebraic representation, particular solutions with different
initial points may have different domains. Determining the proper domain is an
important part of finding the particular solution.
In §A of Chapter 8 we solved more differential equations involving motion
along a straight line. In §B we found parametric equations for the motion of a
particle along a plane curve, given d.e.’s for the components of its acceleration
and velocity.