- E16
[1RK] Discuss the differential equation
for
, studying in a qualitative way the existence (local or global) of solutions, and the properties of monotonicity and convexity/concavity. 1Show that the solution exists for all positive times.
Show that for
the solution does not extend to all negative times.Difficulty:*.Show that there is a critical
such that, for the solution does not extend to all negative times, while for the solution exists for all negative times; also for you have .In dotted purple the line of inflections. In dashed red the parabola where the derivative of the solution is infinite. In yellowthe solutions with initial data
, , .Figure 9 Exercise 5. Solutions for In dotted purple the line of inflections. In dashed red the parabola where the derivative of the solution is infinite. Solutions are drawn with initial data
(”green”), ”orange”)and (”yellow”) . Note that the latter two differ only by in their initial data (indeed they are indistinguishable in the graph for ), but then for they move apart quickly, and for they are respectively and , with a difference of about !Figure 10 Exercise 5. Solutions for Solution 1
EDB — 1RK
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English
Authors:
"Mennucci , Andrea C. G."
.
Bibliography
Book index
- [2] Emilio Acerbi, Luciano Modica, and Sergio Spagnolo. Problemi scelti di Analisi Matematica II. Liguori Editore, 1986. ISBN 88-207-1484-1.
Book index
- ODE
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