Find the particular solution that satifies the differential equation and the initial condition. f" (x) = x2 f' (0) = 8, f (0) = 4

f (x) = x^4/12 + 8x + 4

Step-by-step explanation:

f" (x) = x^2

Integrate both sides with respect to x

f' (x) = ∫ x^2 dx

= (x^2+1) / 2+1

= (x^3) / 3 + C

f (0) = 8

Put X = 0

f' (0) = 0 + C

8 = 0 + C

C = 8

f' (x) = x^3/3 + 8

Integrate f (x) again with respect to x

f (x) = ∫ (x^3 / 3) + 8 dx

= ∫ x^3 / 3 dx + ∫8dx

= x^ (3+1) / 3 (3+1) + 8x + D

= x^4/12 + 8x + D

f (0) = 4

Put X = 0

f (0) = 0 + 0 + D

4 = D

Therefore

f (x) = x^4 / 12 + 8x + 4