How to Solve the Fourth Order Differential Equation D^4 y - 4D^3 y 8D^2 y - 8D y 4 y 0

Understanding and solving higher-order differential equations is a fundamental skill in mathematical modeling and engineering. One such equation is given by:

D^4 y - 4D^3 y 8D^2 y - 8D y 4 y 0

1. Characteristic Equation

The first step in solving a differential equation of the form D^n y a_{n-1}D^{n-1} y ... a_1 D y a_0 y 0 is to find the characteristic equation. For our equation, the characteristic equation is:

r^4 - 4r^3 8r^2 - 8r 4 0

2. Factoring the Characteristic Equation

Although the Rational Root Theorem suggests that this equation has no rational roots, it can be factored into a product of two irreducible quadratics, each of which has a pair of complex conjugate solutions.

The characteristic equation can be factored as:

r^2 - 2r 2^2 0

3. Solving for the Roots

By using the quadratic formula or completing the square, we can solve for the roots:

r 1 ± i

Since these roots are a pair of complex conjugates (1 i and 1 - i) and since they are double roots, the general solution to the differential equation will include these roots in a specific form.

4. General Solution

A general solution to the differential equation is given by:

y e^x [C_1 C_2 x cos{x} C_3 C_4 x sin{x}]

Here, C_1, C_2, C_3, and C_4 are constants determined by initial or boundary conditions.

5. Intuitive Explanation

The presence of complex conjugate roots in the characteristic equation indicates that the solutions of the differential equation include oscillatory behavior (cosine and sine terms) modulated by an exponential term. The double roots imply that the solutions also involve linear terms in these oscillatory components, reflecting the algebraic multiplicity of the roots.

6. Practical Application

This type of differential equation could arise in various applications, such as mechanical vibrations where the system is driven by external forces that can be modeled as oscillatory functions. The solutions can provide insights into the behavior of the system over time, helping engineers and scientists to design systems that can handle such oscillations effectively.

In conclusion, solving higher-order differential equations is a powerful tool in mathematical analysis. Understanding the roots of the characteristic equation and how they affect the form of the solution is crucial for finding solutions to these equations.

Key Points:

Factoring the characteristic equation into irreducible quadratics Solving for complex conjugate double roots Forming the general solution with exponential and oscillatory components