Final answer:
The task involves solving a fourth-order linear homogeneous differential equation with constant coefficients, given specific initial conditions.
Step-by-step explanation:
The question given appears to be related to differential equations with initial conditions. The equation presented looks similar to a fourth-order linear homogeneous differential equation with constant coefficients. The notation y⁽⁴⁾ likely means the fourth derivative of y with respect to x, yᵐ the second derivative of y (commonly noted as y''), and yᵐ again (which appears to be a typo and should refer to y'').
The correct form of the equation should probably be y⁽⁴⁾ - 10y'' + 25y = 0, with the given initial conditions y(0) = 15, y'(0) = 13, y''(0) = 25, y'''(0) = 0. To solve this equation, you would typically look for a solution in the form of y = eᵐˣ, find the characteristic polynomial, and use the initial conditions to determine the constants of the solution.