EDB — 1RK

view in whole PDF view in whole HTML

View

English

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}

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}

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

[1RP]

  1. The differential equation is taken from exercise 13 in [ 2 ] .
Download PDF
Bibliography
  • [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
Managing blob in: Multiple languages
This content is available in: Italian English