Practice question · Multiple choice
A differential equation like dN/dt = kN specifies a rate rather than a formula. Why does one such equation have infinitely many solutions, and what picks out the one you want?
Hints
- Two populations growing at the same percentage rate but starting at different sizes: do both satisfy dN/dt = kN?
- Ask what information the equation does NOT contain.
Show the answer
A. Because the equation says nothing about where N starts.
Why
The equation constrains the shape of the curve and says nothing about its scale, so a colony of 100 and one of 10,000 satisfy it with the same k. The constant is arbitrary in that the equation does not determine it, and it is not free to discard, since C = 0 gives the solution in which nothing exists. It carries the physical information the equation was never going to supply.
Practise Applications of Integration
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