The Euler's Method equation

Aug 2008
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In the Euler's Method section of my textbook, there are two equations mentioned (see attachment). I was wondering if there was any difference between the two, or if they refer to the same thing.

The help would be appreciated (Nod)
 

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HallsofIvy

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Apr 2005
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In the Euler's Method section of my textbook, there are two equations mentioned (see attachment). I was wondering if there was any difference between the two, or if they refer to the same thing.

The help would be appreciated (Nod)
They are different equations the second being more general than the first.

In \(\displaystyle y_{n+1}= y_n+ hF(y_n, t_n)\) we are ignoring the fact that t is a continuous variable and just looking at specific points, \(\displaystyle t_0, y_0)\), \(\displaystyle (t_1, y_1)\), etc. separted by distance h along the t axis.

In \(\displaystyle y(t+ h)= y(t)+ hy'(t)\) we are retaining the continuity of t- here t could be any number. Of course, if we take \(\displaystyle t_0\) to be some specific value, then define \(\displaystyle y_0= y(t_0)\), \(\displaystyle t_1= t_0+ h\), \(\displaystyle y_1= y(t_1)\), \(\displaystyle t_2= t_1+ h= t_0+ 2h\), \(\displaystyle y_2= y(t_2)\), etc. then the second equation becomes the same as the first.
 
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