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Is there any way to rewrite the Laplace transform is such a way that that one can apply to an IVP not centred at zero, that is, at some $y^{(n)}(a_n) = b_n$ for $n\in\mathbb{N}$ and $a_n \in\mathbb{R}\setminus {\{0\}}$, $b_n\in\mathbb{R}$

$\textbf{Edit:}$ For instance, consider the BVP,

$$\frac{d^2y}{dx^2}-3\frac{dy}{dx}+2y=0\quad y(2)=1,y'(4)=8$$

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  • $\begingroup$ Are $\langle a_n \rangle $ and $\langle b_n \rangle$ sequences? Can you give an example? $\endgroup$
    – GFauxPas
    Commented Feb 12, 2015 at 14:59
  • $\begingroup$ @GFauxPas $a_n$ and $b_n$ refer to "points" in a BVP. I just relalised it's bad notation. $\endgroup$
    – user119264
    Commented Feb 12, 2015 at 15:07
  • $\begingroup$ For given conditions $y(t_0)=y_0$ and $y'(t_0)={y_0}'$, you can substitute $v(t)=y(t+t_0)$ and $v'(t)=y'(t+t_0)$ and so on, find the solution for $v(t)$, then solve for $y(t)=v(t-t_0)$. Perhaps you can make an appropriate adjustment for the case that $y(t_0)=y_0$ and $y'(t_1)={y_0}'$? $\endgroup$
    – user170231
    Commented Feb 12, 2015 at 15:35

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