I want to understand the solution sheet to the following question : Solve the cauchy problem: $y' = f(x,y), y(x_0)=y_0$ where $f(x,y) = Ly+ \delta$, $L, \delta \in \mathbb{R}_{>0}$

The solution says there are two possible cases:$ y_0 = -\dfrac{\delta}{L}$ and $y_0 \neq -\dfrac{\delta}{L}$

I understand the second case but for the first case i.e. $ y_0 = -\dfrac{\delta}{L}$ they say the following (which I don't understand):

$f$ is locally Lipschitz continuous in y, $\therefore$ a maximal solution is well defined$

If $ y_0 = -\dfrac{\delta}{L}$ then $y= -\dfrac{\delta}{L}$ is maximal solution.

I don't understand how they came to this conclusion.


$y(x)=-\dfrac{\delta}{L}$ is certainly a solution of the differential equation $y'=L\,y+\delta$ (just plug it into the equation.) Moreover $y(0)=-\dfrac{\delta}{L}$. By uniqueness, it is the unique solution of the equation with initial value $y_0=-\dfrac{\delta}{L}$.


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