Find the particular solution of the differential equation that satisfies the initial condition.
\[
\begin{array}{l}
f^{\prime}(x)=4 x, \quad f(0)=7 \\
f(x)=\square
\end{array}
\]
Answer
Final Answer: \(f(x) = \boxed{2x^2 + 7}\)
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: \(f(x) = \boxed{2x^2 + 7}\)
