Solving ordinary differential equations I(nonstiff problems),exercise 1.1

Solve equation $y'=1-3x+y+x^2+xy$ with another initial value $y(0)=1$.

Solve: We solve this by using Newton's extraordinary method.We assume that the solution is analytic,which means it can be expanded in Taylor series.$y(0)=1$ means that

$$ y'(0)=2 $$ So $$y=1+2x+\cdots$$.

So

$$y'=1-3x+(1+2x+\cdots)+x^2+x(1+2x+\cdots)=2+0\cdot x+\cdots$$

So we have

$$y=1+2x+0\cdot x^{2}+\cdots$$

So we have

$$ y'=1-3x+(1+2x+0\cdot x^{2}+\cdots)+x^2+x(1+2x+0\cdot x^{2}+\cdots)=2+0\cdot x+3x^2+\cdots $$

So we have $$ y=1+2x+0\cdot x^2+x^3+\cdots $$

$$ \vdots $$