Find the particular solution of the differential equation that satisfies the initial condition. \[ \begin{array}{l} f^{\prime}(x)=4 x, \quad f(0)=7 \\ f(x)=\square \end{array} \]

Step 1 ：The given differential equation is a first order linear differential equation.

Step 2 ：The general solution of this differential equation can be found by integrating the right hand side of the equation.

Step 3 ：The general solution of the differential equation is \(f(x) = C1 + 2x^2\).

Step 4 ：Substituting the initial condition \(f(0) = 7\) into the general solution gives the particular solution.

Step 5 ：The particular solution that satisfies the initial condition is \(f(x) = 2x^2 + 7\).

Step 6 ：Final Answer: \(f(x) = \boxed{2x^2 + 7}\)