23.3 Qualitative discussions
For the following exercises the following simple comparison lemma may be useful.
[1R7]Let \(U⊆ ℝ^ 2\) be open, let \(f,g:U→ℝ\) be continuous with \(f≥ g\); let \(I⊆ℝ\) be an open interval with \(t_ 0∈ I\), and let \(x,w:I→ℝ\) solutions of
with \(x(t_ 0)≥ w(t_ 0)\): then \(x(t)≥ w(t)\) for \(t≥ t_ 0\). Note indeed that \(x'(t)≥ w'(t)\) and therefore \(x(t)-w(t)\) is an increasing function.
- E427
[1R8]Discuss solutions of
\[ \begin{cases} y’(x)= (y(x)-x)^ 3\\ y(0)=a~ ~ . \end{cases} \]Qualitatively study the existence (local or global) of solutions, and the properties of monotonicity and convexity/concavity.
Hidden solution: [UNACCESSIBLE UUID ’1R9’][UNACCESSIBLE UUID ’1RB’]
- E427
[1RD] Considering the Cauchy problem
\[ \begin{cases} y’(x) = \frac 1{y(x)^ 2+ x^ 2}\\ y(0)=1 \end{cases} \]show that there is only one global solution \(y:ℝ→ℝ\), that \(y\) is bounded, and the limits \(\lim _{x→∞}y(x)\), \(\lim _{x→-∞}y(x)\) exist and are finite.
![\includegraphics[width=0.9\linewidth ]{UUID/1/R/F/blob_zxx}](images/img-0012.png)
Figure 7 Exercise 2. In purple the line of inflections. In yellow the solutions with initial data \(y(0)=1\) and \(y(0)=2\). Hidden solution: [UNACCESSIBLE UUID ’1RG’][UNACCESSIBLE UUID ’1RH’]
- E427
[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}](images/img-0013.png)
Figure 8 Exercise 3. 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}](images/img-0014.png)
Figure 9 Exercise 3. 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 \(0.7\) !
Hidden solution: [UNACCESSIBLE UUID ’1RP’]
- E427
[1RQ] Note:Exercise 4, written exam 9 July 2011.Show that the Cauchy problem
\[ \begin{cases} y’(x) = y(x)\big( y(x)-x^ 2\big) \\ y(2)=1 \end{cases} \]admits a single solution \(y = y(x)\), defined on all of \(ℝ\) and such that
\[ \lim _{x→−∞} y(x) = +∞ \quad ,\quad \lim _{x→∞} y(x) = 0 \quad . \]