Abstract algebra: systems of differential linear equations

In summary, to solve the inhomogeneous differential equation dX/dt=AX+B in terms of the solutions to the homogeneous equation dX/dt=AX, we can use the general solution of the homogeneous equation X=P-1*Xtilda and a particular solution of the inhomogeneous equation, which can be found using "variation of parameters" with the help of the unknown function u(t).
  • #1
vikkivi
4
0

Homework Statement



Solve the inhomogeneous differential equation dX/dt=AX+B in terms of the solutions to the homogeneous equation dX/dt=AX.


Homework Equations



A is an nxn real or complex matrix and X(t) is an n-dimensional vector-valued function.
If v is an eigenvector for A with eigenvalue a, then X=v*ea*t is a particular solution to the differential equation dX/dt=AX.
And the general solution of the homogenous eqn is X=P-1*Xtilda.


The Attempt at a Solution



So, the general solution of the inhomogeneous equation should be a particular solution of the inhomogenous equations + the general solution of the homogeneous equation. We know the general part, but I am lost on how to find the particular solution for an inhomogeneous equation. Any help would be apprecaited!
 
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  • #2
Is B a constant vector? In that case, you could try a solution where X is also a constant vector, such that dX/dt = 0.
 
  • #3
You can use a multi-dimensional version of "variation of parameters". If the general solution to the homogeneous equation is [itex]X= P^{-1}X^~[/itex] try a solution of the from [itex]Y= P^{-1}X^~u(t)[/itex] where u(t) is an unknown function. Then [itex]Y'= P^{-1}X^~'u+ P^{-1}X^~u'[/itex] and [itex]Ay= AP^{-1}X^~u[/itex] so the equation becomes [itex]P^{-1}X^~'u+ P^{-1}X^~u'+ AP^{-1}X^~u= B[/itex].

Since [itex]X= P^{-1}X^~[/itex] is a solution to the homogeneous equation, [itex]P^{-1}X^~'u+ AP^{-1}X^~u= (P^{-1}X^~'+ AP^{-1}X^~)u= 0[/itex] and the equation reduces to [itex]P^{-1}X^~u'= B[/itex] so [itex]u'= X^~^{-1}PB[/itex]. Integrate that to find u.
 

Related to Abstract algebra: systems of differential linear equations

1. What is abstract algebra?

Abstract algebra is a branch of mathematics that studies algebraic structures such as groups, rings, fields, and vector spaces. It is considered an abstract or generalized form of algebra, as it focuses on the properties and relationships between mathematical structures rather than specific numbers or equations.

2. What are systems of differential linear equations?

A system of differential linear equations is a set of equations that involve multiple functions and their derivatives. These equations can be written in the form of Dx = f(x), where D is a differential operator and f(x) is a given function. These systems are commonly used to model real-world phenomena in the fields of physics, engineering, and economics.

3. How do you solve systems of differential linear equations?

There are several methods for solving systems of differential linear equations, including substitution, elimination, and matrix methods. Depending on the complexity of the system, some methods may be more efficient or accurate than others. It is important to carefully choose the appropriate method for each problem.

4. What is the importance of studying abstract algebra in relation to systems of differential linear equations?

Abstract algebra provides the necessary tools and concepts for understanding and solving systems of differential linear equations. The algebraic structures studied in abstract algebra, such as vector spaces and matrices, are essential in representing and manipulating the functions and variables involved in these systems. Additionally, abstract algebra provides a deeper understanding of the underlying principles and connections between different mathematical structures.

5. What are some real-world applications of abstract algebra and systems of differential linear equations?

Abstract algebra and systems of differential linear equations have numerous applications in various fields, including physics, engineering, economics, and computer science. They are used to model and analyze complex systems, such as electrical circuits, chemical reactions, and population dynamics. Additionally, the concepts and techniques learned in abstract algebra and differential equations are essential for developing advanced technologies, such as artificial intelligence and machine learning algorithms.

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