Solving non-homogeneous system of ODE using matrix exponential

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In summary, the process of solving non-homogeneous systems of ordinary differential equations (ODEs) using matrix exponential involves expressing the system in matrix form, where the non-homogeneous part is treated as an input. The solution is derived by first finding the homogeneous solution using the eigenvalues and eigenvectors of the coefficient matrix, and then employing the matrix exponential to handle the non-homogeneous term. The final solution combines both components, often using the method of variation of parameters or the Laplace transform to account for the non-homogeneous influences. This approach provides a systematic way to tackle complex ODE systems efficiently.
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Homework Statement
Please see below
Relevant Equations
Please see below
For this problem,
1715737007574.png

I don't understand why they include the constants of integration ##c_1 and c_2##, since the formula that we are meant to be using is ##\vec x = e^{At}c + e^{At}\int_{t_0}^{t} e^{-As} F(s)~ds## so we already have the integration variables. Does anybody please know why?

Thanks!
 
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ChiralSuperfields said:
Homework Statement: Please see below
Relevant Equations: Please see below

For this problem,
View attachment 345266
I don't understand why they include the constants of integration ##c_1 and c_2##, since the formula that we are meant to be using is ##\vec x = e^{At}c + e^{At}\int_{t_0}^{t} e^{-As} F(s)~ds## so we already have the integration variables. Does anybody please know why?

Thanks!
Sorry I made a mistake, the equation I'm using is ##\vec x = e^{At}\vec x(0)+ e^{At}\int_{0}^{t} e^{-As} f(s)~ds##
 
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They are evidently using [itex]e^{At}\int^t e^{-As}f(s)\,ds[/itex] with an indefinite integral, so that an arbitrary constant of integration must be included. If instead the integral is made definite with a lower limit of [itex]t_0[/itex], then the arbitrary constant becomes [itex]\vec{x}(t_0)[/itex].
 
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Thank you for your reply @pasmith! Sorry do you mean ##\vec x = e^{At}\vec x(0) + \int e^{At}f(t)~dt## as the indefinite integral?

Thanks!
 
  • #5
No. The indefinite integral already includes an arbitrary constant; you don't need to add an [itex]e^{At}x(0)[/itex] term in this case. That leaves [itex]e^{At} \int e^{-At}f(t)\,dt[/itex].
 
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FAQ: Solving non-homogeneous system of ODE using matrix exponential

What is a non-homogeneous system of ordinary differential equations (ODEs)?

A non-homogeneous system of ordinary differential equations is a system where at least one of the equations includes a term that is not a function of the dependent variable(s) and their derivatives. This term is often referred to as the forcing function or inhomogeneous term. The general form of a non-homogeneous system can be expressed as \( \mathbf{y}'(t) = A \mathbf{y}(t) + \mathbf{f}(t) \), where \( A \) is a matrix of coefficients, \( \mathbf{y}(t) \) is the vector of dependent variables, and \( \mathbf{f}(t) \) represents the non-homogeneous part.

How do you solve a non-homogeneous system of ODEs using the matrix exponential?

To solve a non-homogeneous system of ODEs using the matrix exponential, you first find the general solution to the corresponding homogeneous system \( \mathbf{y}'(t) = A \mathbf{y}(t) \) by calculating the matrix exponential \( e^{At} \). The general solution to the homogeneous part is given by \( \mathbf{y}_h(t) = e^{At} \mathbf{C} \), where \( \mathbf{C} \) is a constant vector determined by initial conditions. Next, you find a particular solution \( \mathbf{y}_p(t) \) to the non-homogeneous equation, which can be done using methods like undetermined coefficients or variation of parameters. The complete solution is then given by \( \mathbf{y}(t) = \mathbf{y}_h(t) + \mathbf{y}_p(t) \).

What is the significance of the matrix exponential in solving ODEs?

The matrix exponential is significant in solving systems of linear ordinary differential equations because it provides a systematic way to express the solution of the homogeneous part of the system. The matrix exponential \( e^{At} \) captures the dynamics of the system and allows for the computation of the state of the system at any time \( t \) based on its initial conditions. It also facilitates the addition of particular solutions to obtain the complete solution of the non-homogeneous system.

What methods can be used to find a particular solution to the non-homogeneous system?

Several methods can be used to find a particular solution to a non-homogeneous system of ODEs, including the method of undetermined coefficients, where you assume a form for the particular solution based on the form of the forcing function \( \mathbf{f}(t) \); the method of variation of parameters, which involves using the solutions of the homogeneous system to construct a

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