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I am trying to understand a bit about solutions of delay differential equations, so I tried analyzing one of the most simple ones: $$u'(t)=-\beta u(t-1), \text{and for } t\in [-1,0), u(t)=\phi(t), \text{for some known function $\phi$}.$$

One way to obtain solutions is to form the characteristic equation by substituting $u(t)=e^{\lambda t}$. Using this method, one can prove that if $\beta=e^{-1}$, then $e^{-t}$ and $te^{-t}$ satisfy $u'(t)=-\beta u(t-1)$. These solutions don't take into account the initial function $\phi$.

Another way of obtaining a solution is to start with the function $\phi$ and integrate the equation in every interval of length $1$. This method is called method of steps.

My question is, what is the relationship between the solutions one obtains from the characteristic equation method and the method of steps?

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If a solution does not take into account a part of the problem (initial condition), then it is not a solution. –  user53153 Jan 23 '13 at 14:32
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