Question

Find the particular solution of the differential equation that satisfies the initial condition(s).

f ''(x) = 3x2, f '(−1) = −3 f(2) = 8

Answer 1

Integrating the given differential equation twice, we get:

f'(x) = x^3 + C1

f(x) = (1/4)x^4 + C1x + C2

Using the initial condition f'(−1) = −3, we get:

-3 = (-1)^3 + C1

C1 = -2

Now the particular solution becomes:

f(x) = (1/4)x^4 - 2x + C2

Using the second initial condition f(2) = 8, we get:

8 = (1/4)(2^4) - 2(2) + C2

C2 = 7

Therefore, the particular solution of the differential equation that satisfies the given initial conditions is:

f(x) = (1/4)x^4 - 2x + 7