Suppose that y(x) is a solution of the autonomous equation dy dx f (y) and is bounded above and below by two consecutive critical points c 1 c 2 , as in subregion R 2 of the following figure If f(y) 0 in the region, then limx y(x) c 2 Discuss why there cannot exist a number L c 2 such that limx y(x...

Solution yx of dydx fy cannot have relative extrema and hence must be monotone Since yx fy ...

Suppose that y(x) is a solution of the autonomous equation dy/dx = f (y) and is bounded

Question:

Suppose that y(x) is a solution of the autonomous equation dy/dx = f (y) and is bounded above and below by two consecutive critical points c₁< c₂, as in subregion R²of the following figure. If f(y) > 0 in the region, then limx †’ ˆž y(x) = c₂. Discuss why there cannot exist a number L < c₂such that limx †’ˆž y(x) = L. As part of your discussion, consider what happens to y'(x) as x †’ˆž

y. R3 y(x) = c2 R2 (x0, Yo) Уx) %3 ст -R1-