Brief Calculus.

Question

Solve the given initial value problem for y = f (x).

dy/dx = 12x² + 3x-5, where y=15 when x = 2

Answer 1

I assume you are looking for a particular solution for the differential equation:

dy/dx = 12x² + 3x-5

∫dy = ∫(12x² + 3x-5)dx

y = 6x³ + (3/2)x² - 5x + C

where y=15 when x = 2:

15 = 6(2)³ + (3/2)(2)² - 5(2) + C

15 = 6(8) + (3/2)(4) - 10 + C

15 = 48 + 6 - 10 + C

15 = 32 + C

C = -17

Therefore the particular solution is:

y = 6x³ + (3/2)x² - 5x - 17