Question

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

F’’(x)=3x^2, f’(-1)=-4, f(2)=5

Answer 1

Integrate the second-order differential equation twice and apply the given initial conditions to solve for the integration constants, resulting in the particular solution.

Step-by-step explanation:

The student is asked to find the particular solution of a second-order differential equation with given initial conditions. To solve this problem, we integrate the equation twice. Given F''(x) = 3x^2, integrating once with respect to x gives us the first derivative F'(x), and integrating again provides us with the function F(x). We then use the initial conditions f'(-1) = -4 and f(2) = 5 to find the constants of integration.

First, we integrate F''(x) = 3x^2 to get F'(x) = x^3 + C1. To find C1, we use the initial condition f'(-1) = -4 which gives C1 = -3. Now we have F'(x) = x^3 - 3.

Next, we integrate F'(x) to get F(x) = (1/4)x^4 - 3x + C2. Using the second initial condition f(2) = 5, we find that C2 = 21/4. Therefore, the particular solution is F(x) = (1/4)x^4 - 3x + 21/4.