Accumulation 6: Differential equations
When students first learn about antiderivatives, they are given simple initial value problems to solve such as . They are instructed to find the antiderivative, then tack on a +C , then substitute in the initial condition, then solve for C and finally write the particular solution. So we hope to see:
But thinking of it as an accumulation problem you can write
Now I’ll admit there is not too much difference in the amount of work involved there, but once the first line is on the paper there is not much to remember in what to do from there on. Common student mistakes in the first method include entirely forgetting the +C and not using the initial condition correctly.
A differential equation in which dy/dx can be written as a function of x only, , with an initial condition has the solution
Look familiar? Yes, that’s my favorite accumulation equation.
Furthermore, if all you need is some function values, this expression can be entered into a graphing calculator and the values calculated without finding an antiderivative.
The scoring standard included a second method that went this way:
You decide which is easier. If you’re still not sure compare the two methods shown for part (c) on the scoring standard.