Find (a) the general solution and (b) the particular solution for the given initial condition. \[ y^{\prime}=\frac{7}{x}+4 x^{2}-x^{4}, y(1)=2 \]

Step 1 ：The given equation is a first order ordinary differential equation. To find the general solution, we need to integrate the right hand side of the equation.

Step 2 ：The general solution of the differential equation is \(y(x) = C1 - \frac{x^5}{5} + \frac{4x^3}{3} + 7\log(x)\).

Step 3 ：To find the particular solution, we need to solve the equation \(y(1) = C1 + \frac{17}{15} = 2\) for \(C1\).

Step 4 ：The particular solution for the initial condition \(y(1) = 2\) is \(y(x) = -\frac{x^5}{5} + \frac{4x^3}{3} + 7\log(x) + \frac{13}{15}\).

Step 5 ：\(\boxed{\text{Final Answer: The general solution of the differential equation is } y(x) = C1 - \frac{x^5}{5} + \frac{4x^3}{3} + 7\log(x)\text{. The particular solution for the initial condition } y(1) = 2\text{ is } y(x) = -\frac{x^5}{5} + \frac{4x^3}{3} + 7\log(x) + \frac{13}{15}\text{.}}\)