Euler's Method with Initial Values: y_{1,0}=1 and y_{2,0}=1

[leopard]

$$y_{2,n} + 0.1(2y_{2,n} - y_{1,n} + 4t)$$

If $$y_{1,0}=1$$ and $$y_{2,0}=1$$, what is $$y_{2, 0.1}$$?

1.1 or 1.14?

 

[HallsofIvy]

Your notation is a bit confusing. Normally an index "n" refers to integer values. What is it here? In other words, what does y_2,0.1 mean? Also there is no equation. What is that expression supposed to be equal to? And, finally, what is "t" here?

 

[leopard]

x'' - 2x' + x = 4t, x(0) = 1, x'(0)=1

Introducing $$y_1 = x$$ and $$y_2 = x'$$

we have the system

$$y_1 ' = y_2$$

$$y_2 ' = 2y_2 - y_1 + 4t$$

right?

Using Euler's method with step size h=0.1, we get

$$y_{1, n+1} = y_{1, n} + 0.1y_{2,n}$$

$$y_{2,n + 1} + 0.1(2y_{2,n} - y_{1,n} + 4t)$$

It's easy to see that $$y_{1, 1} = 1 + 0.1 \cdot 1 = 1.1$$, but what is $$y_{2,1}$$?