Solving non-homogeneous system of ODE using matrix exponential

[ChiralSuperfields]

Homework Statement

Please see below

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!

 

[ChiralSuperfields]

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 [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].

 

[ChiralSuperfields]

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 [itex]e^{At}x(0)[/itex] term in this case. That leaves [itex]e^{At} \int e^{-At}f(t)\,dt[/itex].