Mixed practice studio

Chapter 11 - The method is part of the problem

This studio removes chapter labels. Your first written line for every problem must name a structural feature, a qualitative prediction, or both. Only then should you calculate.

How to use the studio

For each problem:

1. cover the solution link;
2. write a method and confidence from 0 to 100%;
3. solve or analyze;
4. verify with a different representation; and
5. if you miss, record the first wrong decision, not merely the final wrong answer.

Round A: recognize and predict

Do not solve these. Name the most useful first method and one behavior or checking strategy.

1. Mixed - recognition

   M1 \(y'=ty+\sin t\)

   Detailed solution

2. Mixed - recognition

   M2 \(y'=e^t/(1+y^2)\)

   Detailed solution

3. Mixed - recognition

   M3 \(y''+6y'+9y=t\)

   Detailed solution

4. Mixed - recognition

   M4 \(\mathbf{x}'=\begin{bmatrix}0&-1\\1&0\end{bmatrix}\mathbf{x}\)

   Detailed solution

5. Mixed - qualitative

   M5 \(y'=(y-1)(y+2)\)

   Detailed solution

6. Mixed - verification

   M6 A student claims \(y=1/(1-t)\) solves \(y'=y\), \(y(0)=1\). Diagnose the first mismatch.

   Detailed solution

Round B: execute and verify

1. Mixed - first order

   M7 Solve \(y'=(1+t)y^2\), \(y(0)=-1\). State the interval around zero on which the denominator remains nonzero.

   Detailed solution

2. Mixed - first order

   M8 Solve \(y'+(2/t)y=t^2\), \(t>0\), \(y(1)=1\).

   Detailed solution

3. Mixed - second order

   M9 Solve \(y''+2y'+2y=0\), \(y(0)=1\), \(y'(0)=0\). Describe its phase portrait.

   Detailed solution

4. Mixed - forcing

   M10 Find the general solution of \(y''+y=\sin3t\).

   Detailed solution

5. Mixed - numerical

   M11 Use two Euler steps of size \(0.5\) for \(y'=y(2-y)\), \(y(0)=0.5\). Check the values against the phase line.

   Detailed solution

6. Mixed - system

   M12 Classify the origin for \(\mathbf{x}'=\begin{bmatrix}-2&1\\1&-2\end{bmatrix}\mathbf{x}\) and identify the late-time direction for generic initial data.

   Detailed solution

Round C: challenge transfers

C1. Design a safe infusion model

A medication concentration (C(t)) in a fixed blood volume (V) receives a constant infusion (I) mg/hour and clears at a rate proportional to concentration, (kC) mg/hour.

1. Build the differential equation with units.
2. Find and classify the equilibrium concentration.
3. Solve for (C(0)=0).
4. Express the time to reach 90% of equilibrium.
5. Explain which parameters change the target and which change only the approach speed.

Detailed solution

C2. Compare two models, not two formulas

Two populations start at (P(0)=1):

$$P_1^{\prime }=P_1,P_2^{\prime }=P_2(1-P_2/10).$$

Compare them through rate laws, direction fields, early-time behavior, long-run behavior, and numerical sensitivity. Find the first nonzero term at which their Taylor expansions around (t=0) differ.

Detailed solution

C3. Reverse-engineer a portrait

Construct a 2x2 linear system with a spiral sink whose trajectories rotate counterclockwise and whose amplitude envelope is (e^{-0.5t}). Give one valid matrix, its eigenvalues, and the corresponding scalar second-order equation for the first component.

Detailed solution

A four-session spacing plan

Do not finish the whole studio in one sitting.

After Day 14, create six new problems by changing coefficients or initial conditions. Mix them before solving so the labels disappear again.

Exit interview

Answer in complete sentences:

1. What do you inspect before choosing a method?
2. Which visual representation would you use for one autonomous state? For two coupled states?
3. How do you distinguish transient and forced behavior?
4. What evidence makes a numerical result credible?
5. Which error recurred in your log, and what cue will interrupt it?

I completed the studio across multiple sessions and wrote cue-specific corrections for my errors.

Next: Study the detailed solutions by comparing first decisions, not by copying final lines.