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Solve the equation:
$$y^{\prime \prime}=x+y-y^{2}, y(0)=-1, y^{\prime}(0)=1$$

Non linear $$\rightarrow$$ Using Taylor Series method at o:

$$y=y(0)+\frac{y^{\prime}(0)}{1 !} x+\frac{y^{\prime \prime}(0)}{2 !} x^{2}+\frac{y^{\prime \prime \prime}(0)}{3 !} x^{3}+\frac{y^{(4)}(0)}{4 !} x^{4}+\cdots \cdots$$

$$y^{\prime \prime}(0)=0+(-1)-(-1)^{2}=-2$$
$$y^{\prime \prime \prime}(0)=\left[x+y-y^{2}\right]=1+y^{\prime}-2 y y^{\prime}=1+1-2(-1)(1)=4$$
$$y^{(4)}(0)=\left[1+y^{\prime}-2 y {y}^{\prime}\right]^{\prime}=y^{\prime \prime}-2 y^{\prime} y^{\prime}-2 y^{\prime \prime} y=-2-2(1)(1)-2(-2)(-1)=-8$$

$$y=-1+x-x^{2}+\frac{2}{3} x^{3}-\frac{1}{3} x^{4}+\cdots$$

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