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E16

[1RK] Discuss the differential equation

{y(x)=1y(x)x2y(0)=a

for a0, 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>0 the solution does not extend to all negative times.

Difficulty:*.Show that there is a critical a~<0 such that, for a~<a<0 the solution does not extend to all negative times, while for aa~ the solution exists for all negative times; also for a=a~ you have limxy(x)x2=0.

\includegraphics[width=0.9\linewidth ]{UUID/1/R/M/blob_zxx}

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 y(0)=2, y(0)=1, y(0)=1/1000.

Figure 9 Exercise 5. Solutions for a>0

\includegraphics[width=0.9\linewidth ]{UUID/1/R/N/blob_zxx}

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 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>1), but then for x<1 they move apart quickly, and for x=2 they are respectively 3.25696 and 2.54856, with a difference of about 0.7 !

Figure 10 Exercise 5. Solutions for a<0

Solution 1

[1RP]

  1. The differential equation is taken from exercise 13 in [ 2 ] .
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Bibliography
  • [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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