Final answer:
The solution to the differential equation y⁴ + 3y'' - 4y = 0 involves finding the characteristic equation, which is similar to a quadratic equation. The equation is solved by assuming an exponential solution y = e^(rt) and finding the roots for r.
Step-by-step explanation:
The differential equation mentioned, y⁴ + 3y'' - 4y = 0, is a linear homogeneous differential equation with constant coefficients. This can be solved by finding the characteristic equation which is typically in the form r⁴ + 3r² - 4 = 0. We assume a solution of the form y = ert, where r is a constant. The characteristic equation can then be solved using factorization or other algebraic techniques to find the general solution of the differential equation.
To solve the characteristic equation, we look for values of r that satisfy the equation. These solutions for r represent different modes of the system's response and can be used to construct the general solution of the differential equation. This equation is not a quadratic; however, it resembles a quadratic in form and can be treated similarly by substituting r² for a new variable s, thus transforming it into s² + 3s - 4 = 0.