for euler\'s method...for dy/dx=f(x,y) and y(x0)=y0, we start at x0,y0 and take

Question

for euler's method...for dy/dx=f(x,y) and y(x0)=y0, we start at x0,y0 and take the slope f(x0,y0) to use to approximate the next point y(x1)...but we supposedly end the first line at x1,y1...my question is ...if we end the line at x1,y1 then f(x1,y1) must equal f(x0,y0) which doesn't make sense given the circumstances....but if we start a new line at x1,y1 then it still doesn't make sense because the equation y1=y0 + hf(x0,y0) would imply that the difference between y1 and y0 is just h* the slope at x0,y0 which can't be the case if at x1,y1 there is a different slope im basically asking for the polygonal broken line method of approximating the solution...is the next point the end of the first line or the start of the new line...it cannot be both

Explanation / Answer

1) first of all u need to understand what euler's theorom does it actually calculate slope of a curve at a point by taking a very small line such that h tends to 0 now if we do it exactly opposite like y1=y0+h*f(x0,y0) . it makes sense , i this case we are taking x0,y0 as first point and calculating next point using the slope at x0,y0 as h is very small the error in calculating y1 and x1 can be neglected . but if u are using it in a curve which is giving a large erroe then surely it does not makes a sense to use it there 2) euler's fomrmula assumes that x0, y0 and x1 ,y1 are so close , so that the slope change b/w them can be neglected 3) actually you r correct ! but the fact is the differece b/w points can be neglected please rate me

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