u'_1 = x*y_1 - u_2 , u_1(o)=1
u'_2 = xu^2_1 - u_2, u_2(o)=1
Using picard iteration, show that the soln can be obtained for |x|<o.3
Do you find u_1 and u_2 through picard iteration first?
Recall that if $\displaystyle \frac{dy}{dx} = f(x,y),\;\; y(a) = b$ then the picard iterates are obtained by solving
$\displaystyle y_{n+1} (x) = b + \int_a^x f(s,y_n(s))\,ds
,
,y_0 = b$
For systems, this extends and for your system (I'll use x(t) and y(t) b/c of the subscripts)
$\displaystyle x_{n+1}(t) = 1 + \int_0^t s x_n(s) - y_n(s)\, ds
$
$\displaystyle y_{n+1}(t) = 1 + \int_0^t s x_n^2(s) - y_n(s)\, ds$
with $\displaystyle x_0 = 1,\;\;y_0 = 1$. Then calculate the iterates recursively.