How do you begin to show that $f(t,y)$ satisfies a Lipschitz Condition on $[0,1]$ for $y' = f(t,y) = t\sin(\frac12t\pi)$, $y(0) = 1$? There’s no $y$ in the $f(t,y)$, where do I begin?
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3$f$ satisfying the Lipschitz condition in $y$ means that we have $\lVert f(t,y_1)-f(t,y_2)\rVert\leq C\lVert y_1-y_2\rVert$ for some constant $C$. But since there is no $y$-dependence in the left-hand side, it is always zero, so it's trivially true – Lorago Nov 08 '23 at 20:20