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Variation of Parameters p.190 theorem 3.6.1

Variation of Parameters p.190 theorem 3.6.1
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Jeong Yeon Yook

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Variation of Parameters
« on: February 11, 2013, 10:46:40 PM »
In equation (28) of theorem 3.6.1, it says that "where t0 is any conveniently chosen point in I".

This is for the integral limits for u1 and u2.
I don't understand how we can just conveniently choose some t0 for the general solution when it's not an initial value problem.

Victor Ivrii

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Re: Variation of Parameters
« Reply #1 on: February 12, 2013, 02:45:27 AM »
In equation (28) of theorem 3.6.1, it says that "where t0 is any conveniently chosen point in I".

This is for the integral limits for u1 and u2.
I don't understand how we can just conveniently choose some t0 for the general solution when it's not an initial value problem.

Looking for a general solution we can select initial point as we please; it definitely affects other constants. Look at
\begin{equation*}
\int_{t_0}^f f(t)\,dt'+C_0=\int_{t_1}^f f(t)\,dt'+C_1
\end{equation*}
as long as $C_1-C_0=-\int_{t_0}^{t_1}f(t')\,dt'$ and you can select any initial point $t_0$.