Find (a) the general solution and (b) the particular solution for the given initial condition. \[ y^{\prime}=5 x^{4} ; y(0)=8 \]

Step 1 ：The given differential equation is \(y^{\prime}=5 x^{4}\).

Step 2 ：To find the general solution, we integrate the right hand side of the equation. The integral of \(5x^4\) with respect to \(x\) is \(x^5\). Therefore, the general solution is \(y(x) = C1 + x^5\).

Step 3 ：We are given the initial condition \(y(0)=8\). To find the particular solution, we substitute this initial condition into the general solution.

Step 4 ：Substituting \(x = 0\) and \(y = 8\) into the general solution gives us \(8 = C1 + 0^5\). Solving for \(C1\), we find that \(C1 = 8\).

Step 5 ：Substituting \(C1 = 8\) back into the general solution gives us the particular solution \(y(x) = x^5 + 8\).

Step 6 ：\(\boxed{\text{Final Answer: The general solution of the differential equation is } y(x) = C1 + x^5. \text{ The particular solution for the initial condition } y(0) = 8 \text{ is } y(x) = x^5 + 8.}\)