Solve the differential equation

f"(x)=x^2, f'(0)=3, f(0)=-1

f" = x^2

f' = 1/3 x^3 + c
f'(0)=3, so c=3 and
f' = 1/3 x^3 + 3

Now do that again going from f' to f.

To solve the given differential equation, we will use the method of integration.

Step 1: Integrate the given differential equation f''(x) = x^2 with respect to x.
∫f''(x) dx = ∫x^2 dx

Using the power rule of integration, we have:
f'(x) = (1/3)x^3 + C1

Step 2: Apply the initial condition f'(0) = 3 to find the value of the constant C1.
f'(0) = (1/3)(0)^3 + C1
3 = 0 + C1
C1 = 3

Therefore, the equation becomes:
f'(x) = (1/3)x^3 + 3

Step 3: Integrate the above equation f'(x) = (1/3)x^3 + 3 with respect to x.
∫f'(x) dx = ∫[(1/3)x^3 + 3] dx

Using the power rule of integration, we have:
f(x) = (1/12)x^4 + 3x + C2

Step 4: Apply the initial condition f(0) = -1 to find the value of the constant C2.
f(0) = (1/12)(0)^4 + 3(0) + C2
-1 = 0 + 0 + C2
C2 = -1

Therefore, the final solution to the given differential equation f''(x) = x^2 with the initial conditions f'(0) = 3 and f(0) = -1 is:
f(x) = (1/12)x^4 + 3x - 1.