Past exam of the differential equations 1 course of the University of Oxford / 2017 / 2 / b

Where $y\ne -1$, $y^{\prime }=-\sqrt{|1+y|}<0$. By the mean value theorem, the solution is strictly decreasing on every interval where it avoids $-1$. At $y=-1$, the derivative is zero, so a constant segment there is possible. Thus all solutions are nonincreasing, while strict decrease need not persist through an interval spent at that equilibrium.

For example, choose $h=1/4$ and $k=1/2$ around $(a,b)=(0,0)$. On the rectangle, $1+y\ge 1/2$, so $f=-\sqrt{1+y}$ is continuous, $|f|\le \sqrt{3/2}$ and $|\partial f/\partial y|\le 1/\sqrt{2}$. The mean value theorem therefore supplies a Lipschitz constant $L=1/\sqrt{2}$. These choices even satisfy $Mh<k$ and $Lh<1$.

While $y>-1$, separation of variables gives $2\sqrt{1+y}=2-x$. Therefore the uniquely determined branch up to its first arrival at $-1$ isIt reaches $y=-1$ at $x=2$, with derivative zero. The restriction $x\le 2$ is essential: extending the same quadratic beyond two would make its derivative positive and violate the original differential equation. Beyond that time the waiting-time continuation at a non-Lipschitz equilibrium is nonunique.

Two solutions for $x\ge 2$ areFor the second, $y_2^{\prime }=-(x-2)/2=-\sqrt{|1+y_2|}$ because $x\ge 2$; both have value $-1$ and derivative zero at two. They are distinct for every $x>2$. Local Lipschitz continuity fails at the equilibrium, so the existence of these two continuations is compatible with the Picard-Lindelöf theorem.

For each departure time $c\in [2,h]$, defineEach branch satisfies the ordinary differential equation, and both joins have matching value $-1$ and matching derivative zero. Thus $y_c$ is continuously differentiable and satisfies the same initial value problem. If $2\le c_1<c_2\le h$, the solutions differ on $(c_1,c_2)$, proving infinitely many solutions on every $[0,h]$ with $h>2$.

The uniqueness hypotheses fail when the solution reaches $-1$. IndeedNo finite Lipschitz constant works on a rectangle containing that level in its interior. The local theorem still controls the earlier branch near zero; it cannot be extended through a point where its hypotheses fail. These are waiting-time continuations at a non-Lipschitz equilibrium.