Math core problem solutions

Solution to Problem 1 – Triadic sequence scaling#

The sequence is

$$ a_n = D_3^n τ_r. $$

If $$τ_r$$ triples:

$$ τ_r' = 3τ_r, \quad a_n' = D_3^n (3τ_r) = 3a_n. $$

Answer: Each term of the sequence triples.

Solution to Problem 2 – Resonant integral#

$$ I = \int_0^{τ_r} T_f D_6 , dt. $$

Since $$T_f D_6$$ is constant with respect to $$t$$:

$$ I = T_f D_6 \int_0^{τ_r} dt = T_f D_6 τ_r. $$

Answer: $$I = T_f D_6 τ_r$$.

Solution to Problem 3 – Exponential resonance equation#

Solve

$$ X e^{ΛΘ t} = D_9. $$

Divide both sides by $$X$$:

$$ e^{ΛΘ t} = \frac{D_9}{X}. $$

Take logs:

$$ ΛΘ t = \ln\left(\frac{D_9}{X}\right). $$

Thus,

$$ t = \frac{1}{ΛΘ} \ln\left(\frac{D_9}{X}\right). $$

Answer: $$t = \dfrac{1}{ΛΘ} \ln!\left(\dfrac{D_9}{X}\right)$$.