in nature, the strong eat the weak, but in every major religious tradition, the strong are supposed to help the weak. let's try this again today.
divide these 4 problems up and put up your work on the board to share once sufficient progress has been made. problems 5.6.17 a) and b) can check against each other by comparing A(3).
5.6.17: mass-spring-dashpot system
m x" + c x' + k x = F0 cos ωt
m = 1, c = 6, k =45, F0 = 50
a) Investigate the possibility of practical resonance of this system. In
particular find the amplitude A(ω) of steady periodic forced oscillations with
frequency ω. Sketch the graph of A(ω) and find the practical resonance frequency
ω (if any).
b) [added by bob]
For the specific frequency ω = 3, and initial conditions x(0) = 0,
x'(0) = -3, find the steady state solution (particular solution) and the
homogeneous solution (transient solution) and then solve the initial value
problem. Find the amplitude of the steady state solution. Does it agree with the
value A(3) found above? Does it agree visually with a plot of the steady state
solution?
Express the steady state solution in phase-shifted cosine form. Does this look
right compared to your plot?
5.5.20:
y''' - y = exp(x) +7
Find a particular solution yp [Hint: you must find the homogeneous
solution first to decide on the form of your particular solution up to the
undetermined coefficients.]
> deq:=diff(y(x),x,x,x) -y(x)=exp(x)+7;
> dsolve(deq,y(x));
The first two terms give the desired answer to check your derivation.
7.3.11:
x1' = x1 - 2 x2, x2' = 2 x1 + x2, x1(0) = 0, x2(0) = 4.
solve with the eigenvector approach (complex eigenvectors).
[You can check your answer by editing the first order
dsolve system template:
> deqs:=diff(x1(t),t)=x2(t),
diff(x2(t),t)=x1(t);
> inits:=x1(0)=1,x2(0)=2;
> dsolve({deqs},{x1(t),x2(t)});
> dsolve({deqs,inits},{x1(t),x2(t)});
]