Math extended problems (resonance framework)

Problem 4 – Multi-triad polynomial#

Define a polynomial

$$ P(x) = D_3 x^2 + D_6 τ_r x + X. $$

1. Compute $$P(τ_r)$$.
2. If $$τ_r$$ doubles, how does the linear term change?

Problem 5 – Resonant differential equation#

Solve the differential equation

$$ \frac{dy}{dt} = D_6 T_f y, $$

with initial condition $$y(0) = y_0$$.

Problem 6 – Triadic Fourier-like transform#

A signal is defined as

$$ s(t) = e^{-D_3 t}. $$

A triadic transform is defined as

$$ \mathcal{T}{s}(ω) = \int_0^\infty e^{-D_3 t} e^{-i ω τ_r t} dt. $$

1. Write the integrand as a single exponential.
2. Evaluate the integral.

Problem 7 – Resonant fixed point#

Consider the function

$$ f(x) = X \sqrt{x} - D_9. $$

A fixed point satisfies $$f(x) = x$$.
Solve for $$x$$ in terms of $$X$$ and $$D_9$$.

Problem 8 – Triadic matrix resonance#

Let

$$ M = \begin{pmatrix} D_3 & τ_r \ 0 & D_6 \end{pmatrix}. $$

1. Compute $$\det(M)$$.
2. Compute the eigenvalues of $$M$$.