Please answer the second question For both of these situations, you will need to run simulations in SageMath or MATLAB

When last we heard from Romeo and Juliet, they were involved in a love affair modeled by the two-dimensional ODE, R' = J and J' = -R. Recall that Romeo loves Juliet more the more she loves him but that Juliet only starts to like Romeo if he's disinterested in her. We learned that Romeo and Juliet are trapped in a never ending \make-up/break-up" scenario. Romeo's best friend, Mercutio, thinks that Romeo is too good for Juliet, however, and has decided to try get them to break up for good. And so he has started telling Romeo how awful Juliet is. Romeo trusts Mercutio and so his ardor for Juliet wanes a bit when Mercutio tells him such things, though he still really likes Juliet. On the other hand, Juliet really hates Mercutio's guts and the more he disapproves of her relationship with Romeo, the more she likes Romeo! Let R(t) = Romeo's love/hate for Juliet at time t, J(t) = Juliet's love/hate for Romeo at time t, and M(t) = Mercutio's disapproval of Romeo and Juliet's relationship at time t. Positive values of R, J signify love, negative values signify hate. A positive value of M(t) signifies disapproval. Then a model for this complicated saga is:

R'=J-2M

J'=-R+4M

M'=R+4J

1. Using your knowledge of second order linear ODEs, describe what happens to Romeo and Juliet's relationship now. Start with initial values of R=-1, J=1, M=1. USE SAGEMATH.
2. In particular, does Mercutio's tampering have the effect he wants? SAGEMATH
3. And do Romeo and Juliet still oscillate between love and hate? SAGEMATH
4. Sometimes you made need to decrease the end point so you can see what is going on around (0, 0).

#Romeo, Juliet and Mercutio t, x, y, z = var ("t, x, y, z") P = desolve_system_rk4 ([y-2*z,-x+4*z,x+4*y], [x, y, z], ics = [0, 1, 1, 1], ivar = t, end_points=30) Px= list_plot ([[t,x] for t,x,y,z in P], plotjoined=True, legend_label='Romeos Love for Juliet') Py= list_plot ([[t,y] for t,x,y,z in P], plotjoined=True, rgbcolor='red', legend_label='Juliets Love for Romeo') Pz= list_plot ([[t,z] for t,x,y,z in P], plotjoined=True, rgbcolor='black', legend_label='Mercutios Disapproval') show (Px+Py+Pz)