Find a particular solution to the IVP
y' = 4 and y(1) = 9.
We know any solution to the d.e.
y' = 4
has the form
y = 4x + C.
If we're to have y(1) = 9, then
9 = 4(1) + C
and so C has to equal 5. There's no choice. This means the specific solution to this IVP is
y = 4x + 5.
The usual process for solving IVPs is to
We'll be able to solve more kinds of differential equations later, but we can do some practice ones now by thinking backwards.