Taylor series expansion of the solution to a first-order ODE using successive differentiation (Picard/Taylor series method).
Given y'=1+xy, y(0)=1. Differentiating successively and evaluating at x=0: y'(0)=1+0(1)=1. y''=y+xy' so y''(0)=y(0)=1. y'''=2y'+xy'' so y'''(0)=2y'(0)=2. y''''=3y''+xy''' so y''''(0)=3y''(0)=3. y'''''=4y'''+xy'''' so y'''''(0)=4y'''(0)=8. The Taylor series...
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