- E4
[1RK] Discuss the differential equation
\[ \begin{cases} y’(x)=\frac 1{y(x)-x^ 2}\\ y(0)=a \end{cases} \]for \(a≠ 0\), studying in a qualitative way the existence (local or global) of solutions, and the properties of monotonicity and convexity/concavity. 1
Show that the solution exists for all positive times.
Show that for \(a{\gt}0\) the solution does not extend to all negative times.
Difficulty:*.Show that there is a critical \(\tilde a{\lt}0\) such that, for \(\tilde a{\lt}a{\lt}0\) the solution does not extend to all negative times, while for \(a≤ \tilde a\) the solution exists for all negative times; also for \(a=\tilde a\) you have \(\lim _{x→-∞} y(x)-x^ 2=0\).
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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