Approximate Solution for Numerical ODE: Taylor Series Method with Non-Zero Terms

[hazeleyes]

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

 

[lippyka]

The second attempt is correct. The Taylor series expansion of the given equation, with the initial condition y(0)=1, up to the fourth non-zero term is: y(x0+h) = 1 + h + h2/2 + 2h3/6 + 5h4/24. At x=0.1, this gives y(0.1) = 1 + 0.1 + 0.01/2 + 0.002/6 + 0.0005/24 = 1.10042.