Existence interval for ODE

Dear Sir or Madam,
I would be grateful if you can help me to solve the the below question. The system is non-linear, I guess maybe the Picard-lindelof theorem will be helpful for this question.

Given the system
$y'_1=xy_1-y_2,\ y_1(0)=1$
$y'_2=xy^2_1-y_2,\ y_2(0)=1$
Show that the solution can be obtained for $|x|<0.3$ by successive
approximations.