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( a ) Consider the modified problem d 2 y d t 2 + 2 y 2 d y d t + 4 y =

(a) Consider the modified problem
d2ydt2+2y2dydt+4y=3sint, with ,y(0)=-1,dydt(0)=0.
The ODE (7) is very similar to (4) except for the y2 term in the left-hand side. Because of the
factor y2 the ODE (7) is nonlinear, while (4) is linear. There is however very little to change
in the implementation of (4) to solve (7). In fact, the only thing that needs to be modified is
the ODE definition.
Modify the function defining the ODE in LAB04ex1.m. Call the revised file LAB04ex2.m. The
new function M-file should reproduce the pictures in Fig 8.
Include in your report the changes you made to LAB04ex1.m to obtain LAB04ex2.m.
Figure 8: Time series y=y(t) and v=v(t)=y'(t)(left), and phase plot v=y' vs.y for (7).
(b) Compare the output of Figs 7 and 8. Describe the changes in the behavior of the solution in
the short term.
(c) Compare the long term behavior of both problems (4) and (7), in particular the amplitude of
oscillations.
(d) Modify LAB04ex2.m so that it solves (7) using Euler's method with N=400 in the interval
0t55(use the file euler.m from LAB 3 to implement Euler's method; do not delete the
lines that implement ode45). Let [te,Ye] be the output of euler, and note that Ye is a ma-
trix with two columns from which the Euler's approximation to y(t) must be extracted. Plot
the approximation to the solution y(t) computed by ode45(in black) and the approximation
computed by euler (in red) in the same window (you do not need to plot v(t) nor the phase
plot). Are the solutions identical? Comment. What happens if we increase the value of N?
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