Initial value problems

Question

Solve the following initial value problems

a) f'(x) = 5e^x - 3sin(x), f(0) = 0

b) v(t) = 3t⁵ -11t + 9, s(1) = 5

v(t) is the velocity , s(t) is its position

Answer 1

a)

f(x) = ∫f'(x) dx = ∫(5e^x - 3sin x) dx = 5e^x+ 3cosx + c

Since f(0) = 0, we can find c as:

f(0) = 5+3 +c = 8+c = 0, thus c = -8, and then: f(x) = 5e^x+ 3cosx -8

b)

Since the velocity is the derivative of the position:

v(t) = ds(t)/dt

Then: s(t) = ∫ v(t) dt = ∫ (3t⁵-11t + 9) dt

Thus s(t) = 3t⁶/6 -11t²/2 + 9t +c = t⁶/2 - 11/2 t² +9t + c

Since s(1) = 5, we can find c:

s(1) = 1/2-11/2+9+c = 4+c = 5, and thus c = 1

Thus s(t) = t⁶/2 - 11/2 t² +9t + 1