I have to make an assignment on differential equations and Romeo and Juliet.
$r(t)$ is romeo's love for Juliet at time $t$, $j(t)$ is Juliet's love for Romeo at time $t$
So far, it is given: $\frac{dr}{dt}=-j$ and $\frac{dj}{dt}=r$.
It is also given that Romeo & Juliet's families are enemies, thus the initial condition at time $t=0$ is $(r,j)=(-1,-1)$
If we would take the second derivative of $r$ we get: $r''=-j'$. We know that $j'=r$, which means $r''=-r$. can be recognized as the equation of an harmonic oscillator. Our solution will therefore have this shape: $r=A \sin(t)+B \cos(t)$.
To get the solution to $j$, we know $j=-r'$, which gives us:
$Rj= -(A\cos(t)-B\sin(t))= -A\cos(t)+B\sin(t)R$
With the initial conditions:
$r(t)= \sin(t)-\cos(t)$ and $j(t)=-\cos(t)-\sin(t)$
Now, the last part of the assignment is:
“In the Spring a young man’s fancy lightly turns to thoughts of love,” says Tennyson. What differential equation concept is best invoked to capture this idea?
A. a forcing term
B. an unstable equilibrium
C. a nonlinear function for $t$
D. none of the above
Could someone help me with this part?
"Spring" is a time of the year. So the best answer is (C), the only one that references t.