Existence and uniqueness of the solution for the Cauchy problem for ODE system.

Question

Please help me to solve the following problem:

I got the system of differential equation $$\dot x_1=2t \sqrt{|x_1|}+\frac{1}{\sqrt{x_2-1}}, \dot x_2=x_1x_2.$$ I also got initial conditions:

1. $(t_0,x_1^0,x_2^0)=(0, 1,2),$
2. $(t_0,x_1^0,x_2^0)=(0, 0,1),$
3. $(t_0,x_1^0,x_2^0)=(0, 0,2).$

For every case I must determine if Cauchy problem for the system has the solution.

In second part of the problem I have to determine for the following points

4. $(t_0,x_1^0,x_2^0)=(0, 2,3),$
5. $(t_0,x_1^0,x_2^0)=(0, 0,1),$
6. $(t_0,x_1^0,x_2^0)=(1, 0,4).$

if the solution for the Cauchy problem is unique.

Unfortunately I have no idea where to start and I was not able to find relevant section in my textbook. Can you please guide me what should I know to solve this problem, so I will attempt to post my solution for your review.

Thanks a lot for your hints and solutions!

Answer 1

The initial conditions 2 and 5 are not allowed since the r.h.s. is not defined on that points.

Indeed, the r.h.s. of the system $f = (f_1, f_2)$ is defined and continuous in $\Omega :=\{(x_1, x_2)\in\mathbb{R}^2:\ x_2 > 1\}$. By Peano's existence theorem, every Cauchy problem with initial data in $\Omega$ has a solution, hence in the cases 1, 3, 4, 6 there exists at least one solution.

For the uniqueness part, the only problem is for $x_1 = 0$, where $f_1$ is not locally Lipschitz continuous, while $f_1$ is of class $C^1$ in $\{x_1>0\}$ and $\{x_1 < 0\}$ (where it is defined).

Hence, for the condition 4 one has local uniqueness.

Uniqueness holds also for condition 6, because one has $\dot{x}_1 (0) > 0$, so that the solution cannot stay on the discontinuity $\{x_1=0\}$.