What Uniqueness Theorem Tells us About Initial Value Problem

Moderate

Assuming that $0 < t_0 <1$, what does the existence and uniqueness theorem tell us about the initial value problem:

$$tx'+\frac{x}{1-t}=\sqrt{1-x^2},\ x(t_0)=x_0$$

Choose the MOST complete answer.

A

For each $x_0$ satisfying $-1\le x_0\le 1$, the initial value problem has a unique solution that exists for all $t$.

B

For each $x_0$ satisfying $-1\le x_0\le 1$, the initial value problem has a unique solution that exists on the interval $-1 < t <1$.

C

For each $x_0$ satisfying $-1 < x_0 < 1$, the initial value problem has a unique solution that exists on an interval of the form $a < t < b$, where $0\le a < t_0 < b\le 1$.

D

For each $x_0$ satisfying $-1 < x_0 < 1$, the initial value problem has a unique solution that exists on the interval $0 < t < 1$.