 Solution

Rewrite the linear equation as $(x^{2}y{)}^{\prime }=0$. The initial value gives the unique solutionThe Picard iteration is its integral equationThe proposed formula holds for $n=0$. If it holds at $n$, termwise integration of the finite sum and $d(\log t)/dt=1/t$ giveThis proves the expression by induction. The exponential series then gives $y_n(x)\to e^{-2\log x}=x^{-2}$. Convergence is also uniform on every compact subinterval of $(0,\infty )$, since $|\log x|$ is bounded there and the exponential majorant series converges. This is Picard iteration for a simple-pole linear differential equation.