Constant of integration

Maple omits the constant of integration and this can lead to errors if you are not careful. Perhaps the most familiar problem where this arises is an object falling under the force of gravity. You know that this is governed by Newton’s Law

$F=ma$

But the acceleration is just the second derivative of position, so you can express this as

$F=m{z}^{″}$

You also know the force of gravity is the gravitational constant times the mass so we get

$mg=m{z}^{″}$

You can now integrate both sides of this equation with respect to time

${\displaystyle \int gdt}={\displaystyle \int z^{″}\left(t\right)dt}$
$gt+C_1=z^{\prime }\left(t\right)$ , and $z^{\prime }\left(t\right)=v(t)$ the velocity

Now you can integrate each side again to find the position vs. time.

$\begin{array}{l}{\displaystyle \int \left(gt+C_1\right)}dt={\displaystyle \int z^{\prime }\left(t\right)dt}\\ \frac{g{t}^2}{2}+C_{1}t+C_2=z(t)\end{array}$

Using Maple to solve this problem requires you to add in the constants of integration.