Lesson 100 — Consolidation Quarter 10 (ODEs)

Classify and solve: $$.

Classify and solve: $$.

Classify and solve: $$.

Classify and solve: $$ (watch for resonance).

Solve $$, $$. What is the stable equilibrium value?

Solve $$, $$. How long until the temperature reaches 40 °C?

Solve $$, $$, $$. Classify the damping.

Apply Euler's method with $$ to $$, $$, from $$ to $$. Compare with $$.

Solve $$.

Solve $$.

Undamped mass-spring: $$ kg, $$ N/m, $$, $$. Find period, amplitude, and phase.

Verify by direct substitution that $$ solves $$.

$$, $$. What is the limit value $$? At what time is $$?

Solve $$. Explain why an $$ factor appears in the solution.

Bernoulli: solve $$ via substitution $$.

Solve $$. (Hint: reduce to 1st order by letting $$.)

Solve $$.

What is the correct technique to solve the logistic equation $$?

The discriminant $$ with both roots negative. What is the behavior of the solution to $$?

Compare the order of convergence of Euler's method and RK4.

Body found at 10:00 with temperature 33 °C. Room at 20 °C, $$ h⁻¹. Estimate the time of death.

Motorcycle suspension: $$ kg, $$ kN/m, $$. Calculate natural frequency, damping coefficient, and classify the regime.

RC circuit with $$ ms, 5 V step source. At what time is $$ V?

Low-pass RC filter, $$ kHz. Calculate the attenuation (in dB) at $$ kHz.

China: 1.4 billion in 2020, growth rate 0.4%/year. Malthus model predicts population in 2050.

Same data as previous exercise, but using logistic model with $$ billion. Compare with Malthus in 2050.

Drug: $$ mg/L, $$ h⁻¹. How long until concentration drops to 5 mg/L?

Tank with 100 L of pure water receives 5 L/min of brine at 2 g/L; perfect mixture leaves at 5 L/min. What is the amount of salt in 30 min?

Investment with continuous contribution: $$, $$. What is the asset value in 30 years?

Nuclear reactor: $$, $$ s, $$. How long until power doubles?

Demonstrate that the integrating factor formula really solves $$. Also show the uniqueness of the solution given an initial value.

Demonstrate that $$ and $$ are linearly independent for $$ by calculating the Wronskian.

Demonstrate that Euler's method has a global convergence order of 1.

Lotka-Volterra. $$, $$. Find equilibria, analyze stability via Jacobian, and estimate the oscillation period near the non-trivial equilibrium.

Large pendulum. $$, $$, $$. Express the exact period as an elliptic integral and compare with the small-angle approximation.