The difference between the differential equations in Activities 2 and 3 is important. In Activity 3, it is the independent variable that appears on the right-hand side. Thus, we are looking for a function whose derivative is `2t`. One such function is `t^2`; another is `t^2+3`. In fact, if `f` is any function of the form `ftext[(]t text[)]=t^2+C`, where `C` is any constant, then `f` satisfies the differential equation `dy text[/]dt=2t` because `f'text[(]t text[)]=2t+0=2t`.
Thus, again we find that the differential equation has infinitely many solutions. Some of the differences between the two families of solutions are described in our discussion of Activity 2. We can also describe the difference in terms of formulas:
If the slope of a function is always proportional to the function, then the function is a constant times an exponential function.
If the slope is always proportional to the independent variable, then the function is a constant plus a quadratic power function.