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I have this weird question from my book that I don't know how to solve. It is suppose that $p(x)$ is a Lipschitz function and $q(x)$ is continuous and then for the equation $x' = p(x)$ and $y' = q(x)y$, the initial value problem has a unique solution.

My confusion about this problem is that the initial value problem is defined to be $x'= f(t,x)$ and $x(t_0) = x_0$ but why is both $x'$ and $y'$ here? And I am not sure what I must show, do I have to show these conditions somehow imply the conditions for the Picard Theorem?

Thanks.

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  • $\begingroup$ Is $g(x)$ supposed to be $q(x)$? $\endgroup$ – Robert Israel Oct 7 '13 at 5:25
  • $\begingroup$ oops, yep [characters to go] $\endgroup$ – user96865 Oct 7 '13 at 5:27
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Presumably the "initial value problem" here is a system of two differential equations: $x' = p(x), y' = q(x) y$, with two initial conditions: $x(t_0) = x_0, y(t_0) = y_0$.

Hint: first consider $x' = p(x)$ with $x(t_0) = x_0$, and show the solution of that is unique. Then look at $y' = q(x) y$ with $y(t_0) = y_0$. In both of those cases you'll apply the Existence and Uniqueness Theorem.

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