Undetermined coefficients - nonhomogenous system

[Heneng]

Hello!

I am taking a course in differential equations and the book I am using is "Elementary Differential Equations" - E. Boyce & R. DiPrima (tenth edition)

My question is about guessing the form of a particular solution to a nonhomogenous system of equations.

Given a nonhomogenous system of differential equations on the form: x' = Px + g(t) where P is a constant n×n - matrix and g(t) is a continuous vector-function for α ∠ t ∠β.
Let υ, ν, and ρ be n×1 - vectors

On page 442 it is said:

"The procedure for choosing the form of the solution is substantially the same as that given in Section 3.5 for linear second order equations.
The main difference is illustrated by the case of a nonhomogenous term of the form υe^λt , where λ is a simple root of the characteristic equation. In this situation, rather than assuming a solution of of the form νte^λt , it is necessary to use νte^λt + ρe^λt , where ν and ρ are determined by substituting into the differential equation."

How did they come up with νte^λt + ρe^λt ? Why is it necessary for systems to do this and not with just a single nonhomogenous equation?

Hope someone can come up with an explanation for this.

 

[jvicens]

Hello,

Thank you for your question and for sharing the book you are using. The procedure for choosing the form of the solution to a nonhomogeneous system of differential equations is based on the method of undetermined coefficients. This method is used to find a particular solution to the nonhomogeneous equation by assuming a form for the solution and then solving for the coefficients.

In the case of a single nonhomogeneous equation, the form of the solution can be assumed to be of the form νteλt, where ν is a constant and λ is a root of the characteristic equation. This is because the single equation can be solved using the method of undetermined coefficients.

However, in the case of a nonhomogeneous system of equations, the form of the solution must take into account the matrix P in the equation. This means that the solution will be a vector function rather than a scalar function. Therefore, the form of the solution must include a vector component, hence the addition of ρeλt to the solution.

The reason for this is that the matrix P acts on the solution, and in order for the solution to satisfy the nonhomogeneous equation, it must have a component that is not affected by P. This is where the vector ρ comes in, as it can be chosen to be unaffected by P. This is why the form of the solution for a nonhomogeneous system of equations is νteλt + ρeλt.

I hope this explanation helps to clarify why this form of the solution is necessary for nonhomogeneous systems of equations. Please let me know if you have any further questions. Best of luck with your studies!