Final answer:
The question appears to involve solving a fourth-order linear differential equation with initial conditions specified, but the provided information contains inconsistencies. A typical approach would use characteristic equations or variation of parameters, but due to the mismatch in initial condition orders, a solution cannot be provided without clarification.
Step-by-step explanation:
The student's question involves solving the initial value problem for a fourth-order linear differential equation. However, there seems to be a mismatch between the information in the question and usual practices for solving differential equations. The equation provided contains a fourth power of the function y (y⁴), but the question also mentions y(0), y'(0), y"(0), and y³(0), which suggests initial conditions for at most a third-order differential equation. It's unclear whether the 'y³(0) = 1' is a typo or an additional piece of information that doesn't fit with the standard form of an initial value problem.
Assuming it is a typo and should be y³(t), the presence of an inconsistent order of derivative is problematic for solving the problem as stated. Therefore, without clarification, it's impossible to provide a direct solution. To solve such a problem under normal circumstances, one would apply techniques such as the characteristic equation method or variation of parameters, given the correct initial values are provided.