- 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 \(\lim _{x→-∞} y(x)-x^ 2=0\).
![\includegraphics[width=0.9\linewidth ]{UUID/1/R/M/blob_zxx}](/UUID/EDB/1RK/html/images/img-0001.png)
Figure 9 Exercise 5. Solutions for \(a{\gt}0\) In purple the line of inflections. In red the parabola where the derivative of the solution is infinite. In yellow the solutions with initial data \(y(0)=2\), \(y(0)=1\), \(y(0)=1/1000\).
![\includegraphics[width=0.9\linewidth ]{UUID/1/R/N/blob_zxx}](/UUID/EDB/1RK/html/images/img-0002.png)
Figure 10 Exercise 5. Solutions for \(a{\lt}0\) In purple the line of inflections. In red the parabola where the derivative of the solution is infinite. Solutions are drawn with initial data \(a=-1.4\) (”green”), \(a=-1.0188\) (”orange”) and \(a=-1.019\) (”yellow”). Note that the latter two differ only by \(0.0002\) in their initial data (indeed they are indistinguishable in the graph for \(x{\gt}-1\)), but then for \(x{\lt}-1\) they move apart quickly, and for \(x=-2\) they are respectively \(3.25696\) and \(2.54856\), with a difference of about \(0.7\) !
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.
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- ODE
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