Final answer:
To solve the initial value problem, the differential equation needs to be integrated twice using the given initial condition y(4) = 2 to find the unique solution.
Step-by-step explanation:
The student's question is focused on solving an initial value problem in the subject of Mathematics. This problem involves a second-order differential equation with initial conditions. To solve this problem, we must integrate the given equation twice and then use the initial condition y(4) = 2 to find constants of integration. Unfortunately, the provided excerpts containing various equations and steps seem not to relate directly to the initial value problem in question and therefore, cannot be used to solve the problem accurately.
The solution begins by writing the differential equation in its standard form and then finding the general solution. The initial value y(4) = 2 is then employed to find the specific solution that satisfies this condition. This requires knowledge of differential equations and methods of solving them, such as characteristic equations for linear homogeneous differential equations, or integrating factors for non-homogeneous equations.