Problem solving with integrals
Answers are for parents and teachers.
1
Find the solution of y′ = 2cos x − 1 with y(0) = 3.
y = 2 sin x − x + C, y(0) = C = 3, so y = 2 sin x − x + 3.
2
If y′ = −2y and y(0) = 7, find y(t).
y = 7e^(−2t).
3
Tea is at 90 °C and the room at 20 °C. The temperature difference halves every 5 minutes. What is the tea’s temperature after 10 minutes?
The difference goes 70 → 35 → 17.5; T = 20 + 17.5 = 37.5 °C.
4
If v(t) = 3t² − 2t and s(0) = 1, find s(2).
s = t³ − t² + 1, so s(2) = 8 − 4 + 1 = 5.
5
Why is an initial condition needed in a concrete differential-equation problem?
The general solution contains a constant C and describes infinitely many processes. The starting state (for example the initial amount or position) fixes C and singles out one real process.