This ODE $(\dot x,\dot y)=(f(x),g(x)y)$ has only one solution

Question

I'm trying to solve this question:

Let $f,g:\mathbb R\to \mathbb R$ be continuous and Lipschitz. Prove the initial value problem:

$(\dot x,\dot y)=(f(x),g(x)y)$, $x(t_0)=x_0$, $y(t_0)=y_0$ has only one solution in every interval.

I've tried to use the Picard theorem without success, I think maybe I have to adapt this theorem for this case.

I really need help

every help is welcome

Thanks in advance

Answer 1

Since the equation for $x$ does not involve $y$, you can solve it first as a single ODE: $$\dot x = f(x), \quad x(t_0)=x_0\tag1$$ Since $f$ is globally Lipschitz, the Picard theorem implies that the solution of (1) exists for all times, and is unique. Now that $x$ is a known function, the equation for $y$ turns into a linear ODE $$\dot y = g(x)y, \quad y(t_0)=y_0\tag2$$ Here $g(x)$ is a continuous function of $t$, being the composition of two continuous functions. It follows that the solution of (2) exists for all times, and is unique.

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Remark #1: "continuous and Lipschitz" is redundant. A Lipschitz function is automatically continuous.

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Remark #2, concerning your meta question. The Tumbleweed badge is not completely useless. It serves as a catalog of overlooked questions, and thus draws some (small) amount of attention toward them -- which is better than nothing.