U^'''-9 u^''+27 u^'-27 u=0

Name: Problem 9, Chapter 6: Fundamentals of Differential Equations
Uploaded: 2020-07-22T20:50:23-08:00
Duration: 1 min 29 s
Channel: Ibrahima Barry
Description: u^'''-9 u^''+27 u^'-27 u=0

Key Concepts

Linear Homogeneous Differential Equations with Constant Coefficients

These are differential equations in which the dependent variable and its derivatives appear linearly, and the coefficients are constants. The general solution is typically found by assuming an exponential form for the solution, which leads to the creation of an auxiliary or characteristic equation that determines the behavior of the solutions.

Characteristic Equation

The characteristic equation is a fundamental tool used to solve linear homogeneous differential equations with constant coefficients. By substituting an exponential trial solution into the differential equation, the differential equation is converted into a polynomial equation (the characteristic equation) whose roots correspond to the exponents in the solution's exponential terms.

Repeated Roots

Repeated roots occur when the characteristic equation has roots with multiplicity greater than one. Repeated roots imply that the standard exponential solution associated with that root must be multiplied by increasing powers of the independent variable to provide a complete set of linearly independent solutions, ensuring the general solution captures the full solution space.

General Solution Structure for Repeated Roots

When the characteristic equation yields a root with multiplicity m, the general solution incorporates terms that are the product of the exponential solution and powers of the independent variable. Specifically, if r is a repeated root of multiplicity m, the solution will include terms like e^(rx), x*e^(rx), x^2*e^(rx), and so on, up to x^(m-1)*e^(rx), thereby forming a complete basis for the solution space.

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