Use method of undetermined coefficients to find Taylor series solution of: $y'+y=2x^2, \space \space \space y(1)=2$

Question

As the title says. The answer is supposedly $$y=2+2(x-1)^2.$$ I can find this answer using other methods however I am unsure of how to use method of undetermined coefficients to find the solution.

Answer 1

Write $y(x)$ in its Taylor expansion around $x=1$,

$$y(x) = y(1)+y'(1)(x-1)+\frac 12 y''(1)(x-1)^2\tag{1}$$

Use $y'=2x^2-y$ and $y''=4x-y'$ to evaluate the required quantities in above expression,

$$y(1) = 2$$ $$y'(1) = 2\cdot 1^2-y(1)=0$$ $$y''(1) = 4\cdot 1-y'(1) = 4$$

Plug above values into (1) to obtain the solution,

$$y(x) = 2+2(x-1)^2$$

Note that $y’’’(1)$ and the higher derivatives are all zero. So, there is no need to go beyond the second order.