 Solution

In position-velocity variables $(u,v)$, the scalar right-hand side isIts only relevant singularity is $u=0$. The initial state $(x,u,v)=(0,1,1)$ is separated from it. For an explicit application of part (a), take $h=1$ and $k=1/2$ in the L1 norm. On the resulting closed ball, $1/2\le u,v\le 3/2$, and $|x|\le 1$.

The partial derivatives obeyThe ball is convex, so integrating these partial derivatives along a segment between two states gives the uniform Lipschitz conditionContinuity holds on the whole slab, andThus $L=9$, $M\le 9$, and the choice $\boxed{\eta =1/20}$ satisfies the self-map and contraction bounds in part (a). This proves a unique solution on $[-1/20,1/20]$, with $y$ bounded away from zero there.

There is also a useful exact representation. Where $y>0$, set $q=y^{\prime }/y$, the logarithmic derivative. Differentiation cancels the quadratic term:Solving this first-order linear differential equation gives $q=e^{x^2}$, and a second integration yieldsThis function is positive and smooth for every real $x$, satisfies both initial data, and verifies the original ordinary differential equation by direct differentiation. It therefore extends the unique local solution to the entire real line. The logarithmic-derivative reduction of a second-order equation explains why this apparently nonlinear example is explicitly integrable by integration.