Solving non-homogeneous system of ODE using matrix exponential

[member 731016]

Homework Statement

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Relevant Equations

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For this problem,

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!

 

[member 731016]

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 $\vec x = e^{At}\vec x(0)+ e^{At}\int_{0}^{t} e^{-As} f(s)~ds$

 

[pasmith]

They are evidently using $e^{At}\int^t e^{-As}f(s)\,ds$ 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 $t_0$, then the arbitrary constant becomes $\vec{x}(t_0)$.

 

[member 731016]

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!

 

[pasmith]

No. The indefinite integral already includes an arbitrary constant; you don't need to add an $e^{At}x(0)$ term in this case. That leaves $e^{At} \int e^{-At}f(t)\,dt$.