Exercises 19.6 Homework
1.
(Theoretical) Rewrite the second-order equation \(y'' + yy' - ty^3 = e^{-t}\) as a system of two first-order equations.
2.
Use the midpoint method (19.4.3) with step size \(h=0.01\) to solve the autonomous system
\begin{align*}
y_1' \amp = y_2\\
y_2' \amp = -y_1 - y_2^3
\end{align*}
with initial condition \(y(0) = \begin{pmatrix}1 \\ 1\end{pmatrix}\) on the interval \([0, 50]\text{.}\) Display both time series plot and phase plot side by side, for example
subplot(1, 2, 1) plot(t, y) subplot(1, 2, 2) plot(y(1,:), y(2,:))
How would you describe the long-term behaviour of this solution?