Question

Solve the differential equation. $$ f^{\prime}(s)=6 s-8 s^{3}, f(2)=3 $$

Short answer

The solution to the differential equation is \(f(s) = 3s^2 - 2s^4 - 7\).

Step-by-step solution

Integrate the function

The first step is to integrate the given function. The integral of \(f'(s)=6s-8s^3\) gives us the original function \(f(s)\). So, \(\int f'(s) \, ds = \int (6s-8s^3) \, ds\). By integrating, one can find that \(f(s) = 3s^2 - 2s^4 + C\), where C is the constant of integration.

Use Initial Condition

Next step is to use the initial condition \(f(2)=3\) to find the constant C. Substituting these values into \(f(s) = 3s^2 - 2s^4 + C\), one gets \(3 = 3(2)^2 - 2(2)^4 + C\). From this equation, it can be found that \(C = -7\), which gives the final solution of our differential equation.

Write the Final Solution

Now we know that the value of C is -7, we can write our final solution. So, the solution to the differential equation is \(f(s) = 3s^2 - 2s^4 - 7\).