hi, I'm wondering how to find the approximate solution of

dy/dx= e^x y y(0)=1

at x = 0.1 using the Taylor Series method. The expansion should include the

first four non-zero terms. Work to six decimal places accuracy.

Here is what I did but I am unsure which one is correct :

first attempt:

y'=e^x y y'(0)=1

y''=e^x y' y''(0)=1

y'''=e^x y'' y'''(0)=1

sub them into Taylor series method:

y(x0+h)=y(0+0.1)=y(0.1)= 1+h+h^2 /2 + h^3 /6

second attempt:

used product rule to differentiate:

y''=e^x y' + e^x y y''(0)=1+1=2

y'''=e^x y'' + e^x y' + e^x y' +e^x y y'''(0)=2+1+1+1=5

sub them into Taylor:

y(0.1)=1+1*0.1+ 2*(0.1)^2 /2 + 5*(0.1)^3 /6

thank you