I need help finding solution to differential equation with taylor approximation.

Question

I have the differential equation

$$y'(t)=y(t)^9-13t^{13}$$

I need to approximate the solution with a 2nd order taylor polynomial in 1 where $y(1)=-1$

So I find $y(1), y'(1)$ and $y''(1)$

$$y(1)=-1$$ $$y'(1)=(-1)^9-13(-1)^{13}=-1+13=12$$

I differentiate $y(t)^9-13t^{13}$ to get $y''(t)$

$$y''(t)=(y(t)^9-13t^{13})'=9y(t)^8y'(t)-169t^{12}$$

and get that

$$y''(1)=9(-1)^8\cdot12-169(-1)^{12}=108-169=-61$$

But when I use Maple to find the solution, I can see that my calculated values are wrong:

Can anyone see where I've gone wrong?

Answer 1

Since $$y'(\color{red}{t})=y(t)^9 - 13\cdot\color{red}{t}^{13}$$ you should have $$y'(1) = (y(1))^9 - 13\color{red}{(+1)}^{13}$$

Answer 2

You inserted $t=1$ wrong, you got for some reason $(-1)^{13}$ instead of $t^{13}=1^{13}=1$. Probably confused it with the value of $y(1)=-1$.