Taylor series of the solution of y′ = x + y²

Successive derivatives.
- $y'(0)=0+1^2=1$.
- Differentiating: $y''=1+2yy'$, so $y''(0)=1+2\cdot1\cdot1=3$.
- Differentiating again: $y'''=2(y')^2+2yy''$, so $y'''(0)=2+6=8$.

Taylor polynomial.
$$y(x)=1+x+\frac32x^2+\frac43x^3+o(x^3).$$

Check. Substitute into the equation: $y'=1+3x+4x^2$ and $x+y^2=x+1+2x+\left(1+3\right)x^2+\cdots=1+3x+4x^2+\cdots$ ✓.

Remark. This equation has no solution in elementary functions (it is a Riccati equation related to Airy functions), yet we get as many coefficients as desired by repeated differentiation. Note also that this solution blows up at a finite $x$, so the series has a finite radius of convergence.