Taylor Method iterative formula, HELP please

Question

dY/dt = 2ty

y(0)=1

a) Determine the order 2 Taylor Method iterative formula for the above equation and calculate y(1) with step h = 0.2.

b)calculate the exact solution of the PVI and the error made in the approximation of the previous item to the value of y(1).

I know the exact solution => y(t)=e^(t^2)

Answer 1

I'll start it off for you:

y_n+1 = y_n + hy'_n + (h²/2)y^''_n

From the ODE: y' = 2ty and y'' = 2y'+2y = 4ty + 2y

y_n+1 = y_n + h(2ty_n) + (h²/2)(4t+2)y_n

y_n+1 = (1+2ht+(h²/2)(4t+2))y_n Last term can be written as h²(2t+1)

Now plug in h = .2 and t incrementing from 0 to .8 (5 iterations) (note y_n is y(t_n))

Here's the first iteration:

y(.2) = (1 +2*.2*0+(.2²/2)(4*0 +2))(1) = 1.02

Second iteration

y(.4) = (1 + 2*.2*.2 +(.2²/2)(4*.2 +2))(1.02) = 1.15872)

and so on...

When you are done, you can calculate error from actual solution at t=1

Please check my math as there are a lot of niggly things.

Please consider a tutor. Take care