Chapter 19 — Exercises

Forty problems on separable equations, first-order linear equations and integrating factors, slope (direction) fields, Euler's method by hand, exponential and logistic growth, mixing, Newton's cooling, radioactive decay, and the SIR model. Each problem is tiered by difficulty: ⭐ (mechanical), ⭐⭐ (standard), ⭐⭐⭐ (multi-step), ⭐⭐⭐⭐ (challenge / synthesis).

A habit worth keeping (§19.2): every solution to an ODE can be checked by substitution. Differentiate your answer, plug it back in, and confirm both sides agree before you trust it.

Part A — Separable Equations (§19.3)

19.1 ⭐ Solve $\dfrac{dy}{dx} = 3x^2$. (A pure antiderivative — the simplest "ODE.")

19.2 ⭐ Solve $\dfrac{dy}{dx} = 2xy$ with $y(0) = 1$.

19.3 ⭐ Solve $\dfrac{dy}{dx} = \dfrac{x}{y}$ with $y(0) = 1$. Confirm your answer is $y = \sqrt{x^2 + 1}$ by substitution.

19.4 ⭐⭐ Solve $\dfrac{dy}{dx} = y\cos x$ with $y(0) = 1$.

19.5 ⭐⭐ Solve $\dfrac{dy}{dx} = e^{x - y}$. (Hint: $e^{x-y} = e^x e^{-y}$.)

19.6 ⭐⭐ Solve $\dfrac{dy}{dx} = y^2$ with $y(0) = 1$. Show that the solution blows up in finite time and find that time — the same finite-time blow-up phenomenon flagged in Worked Example 19.3.2.

19.7 ⭐⭐ Solve $\dfrac{dy}{dx} = (1 + y^2)\tan x$.

19.8 ⭐⭐⭐ Solve $\dfrac{dy}{dx} = \dfrac{1 - y}{x + 1}$ with $y(0) = 0$. Leave the answer explicit in $y$.

Part B — First-Order Linear Equations and Integrating Factors (§19.4)

19.9 ⭐ Solve $y' + y = e^{-x}$. Identify the integrating factor first.

19.10 ⭐⭐ Solve $y' - 2y = e^{3x}$.

19.11 ⭐⭐ Solve $y' + \dfrac{2}{x}\,y = 3x^2$ for $x > 0$.

19.12 ⭐⭐ Solve $x\,y' + y = \cos x$ for $x > 0$. (Notice the left side is already $(xy)'$ — confirm that the integrating factor reproduces this.)

19.13 ⭐⭐⭐ Solve $y' + 2y = e^{-2x}$ with $y(0) = 0$. Explain why the integrating-factor and the answer's structure (homogeneous + particular, §19.4) make the $y(0)=0$ solution especially clean.

19.14 ⭐⭐⭐ Solve $y' \tan x + y = \sin x$ on $(0, \pi)$ with $y(\pi/2) = 1$. (First write it in standard form $y' + P(x)y = Q(x)$.)

Part C — Slope (Direction) Fields (§19.5)

19.15 ⭐ For $y' = x - y$, compute the slope of a solution curve at the points $(0,0)$, $(1,0)$, $(0,1)$, and $(2,1)$. On which line is the slope exactly $0$ (the zero-slope isocline)? (Compare to the worked field in §19.5.)

19.16 ⭐⭐ Sketch (by hand, on a $5\times 5$ grid over $[-2,2]^2$) the direction field of $y' = y$. Describe the family of solution curves you read off the arrows, and confirm it matches $y = Ce^x$.

19.17 ⭐⭐ For the logistic field $P' = 0.5\,P(1 - P/100)$, mark where the arrows are horizontal (slope $0$). Identify the two equilibria and classify each as stable or unstable by the direction the nearby arrows point.

19.18 ⭐⭐⭐ Match three direction fields to three equations without solving: $y' = x$, $y' = y$, $y' = -y$. Justify each match by describing one distinguishing feature of the arrows (e.g., where slopes are zero, where they grow).

Part D — Euler's Method by Hand (§19.6)

19.19 ⭐⭐ Apply Euler's method by hand to $y' = y$, $y(0) = 1$, with $h = 0.5$, for two steps (to $t = 1$). Tabulate $(t_n, y_n)$. Compare $y_2$ to the exact value $e \approx 2.718$ and explain the direction of the error.

19.20 ⭐⭐ Apply Euler by hand to $y' = -2y$, $y(0) = 1$, with $h = 0.25$, for two steps. Compare to the exact $e^{-2t}$ at $t = 0.5$ (§19.6 worked example).

19.21 ⭐⭐⭐ Apply Euler by hand to $y' = x + y$, $y(0) = 1$, with $h = 0.2$, for three steps (to $x = 0.6$). Keep four decimals.

19.22 ⭐⭐⭐ For $y' = y$, $y(0) = 1$, Euler with step $h$ gives $y(1) \approx (1+h)^{1/h}$. Explain why, and use it to show the global error is $O(h)$ (first order). What does $h \to 0$ give, and why is that reassuring?

19.23 ⭐⭐⭐⭐ Euler's method becomes unstable for $y' = -2y$ when $h$ is too large. Using one Euler step, find the threshold value of $h$ above which $|y_{n+1}| > |y_n|$ even though the true solution decays. (Hint: the step multiplies by $1 + hF = 1 - 2h$; see the Computational Note in §19.6.)

Part E — Exponential Growth and Decay (§19.3, §19.10)

19.24 ⭐ A population grows by $P' = 0.05\,P$ with $P(0) = 100$. Find $P(t)$ and the time at which $P = 1000$.

19.25 ⭐⭐ Radioactive decay (nuclear physics). Carbon-14 has half-life $5730$ yr. A bone fragment measures $25\%$ of the living-tissue C-14 level. Estimate its age using $N = N_0 e^{-kt}$, $k = \ln 2 / 5730$.

19.26 ⭐⭐ Drug elimination (pharmacology). A drug clears with half-life $3$ hr from a $200$ mg dose. What amount remains at $12$ hr and at $24$ hr? Express the answer as a multiple of the dose.

19.27 ⭐⭐ Continuous compounding (finance). \$10{,}000 is invested at a $5\%$ nominal annual rate, compounded continuously ($M' = rM$). Find the balance after $30$ years.

Part F — Mixing Problems (§19.4)

19.28 ⭐⭐ Flushing a tank (environmental engineering). A tank holds $200$ L of water with $50$ g of dissolved salt. Pure water flows in at $5$ L/min and the well-stirred mixture drains at $5$ L/min. Set up and solve for $y(t)$, the grams of salt; find the half-life of the salt content.

19.29 ⭐⭐⭐ Brine inflow (chemical engineering). Repeat 19.28 but with inflow of salt water at $1$ g/L (still $5$ L/min in and out). Find $y(t)$ and the steady-state salt content. Compare to the §19.4 worked mixing example.

19.30 ⭐⭐⭐⭐ Changing volume (chemical engineering). A tank starts with $100$ L of pure water. Brine at $2$ g/L flows in at $4$ L/min, but the mixture drains at only $2$ L/min, so the volume rises. Set up the ODE for the salt $y(t)$, being careful that the concentration is $y/V(t)$ with $V(t) = 100 + 2t$ (the trap flagged in the Common Pitfall of §19.4). Solve it. (This is the hardest mixing problem in the set — the integrating factor involves $V(t)$.)

Part G — Newton's Law of Cooling (§19.8)

19.31 ⭐⭐ A $90^\circ$C cup of coffee in a $20^\circ$C room cools to $70^\circ$C in $10$ minutes. When will it reach $30^\circ$C?

19.32 ⭐⭐⭐ Forensics — time of death. A body is found with core temperature $32^\circ$C in a room at $20^\circ$C. One hour later it is $30^\circ$C. Assuming a normal body temperature of $37^\circ$C at death, estimate when death occurred — the method of Worked Example 19.8.1.

19.33 ⭐⭐⭐ Warming (food science). A roast at $80^\circ$C is placed in a $180^\circ$C oven. After $30$ minutes it reaches $100^\circ$C. When will it reach $160^\circ$C? (The cooling law also runs in reverse for heating; only the sign of the initial gap changes — §19.8.)

Part H — Logistic Growth (§19.7)

19.34 ⭐⭐ Solve the logistic IVP $P' = 0.5\,P(1 - P/1000)$, $P(0) = 10$, in closed form using $P(t) = K/(1 + Ae^{-rt})$. At what population is $P$ growing fastest, and at what time does that occur?

19.35 ⭐⭐⭐ A bacterial culture follows logistic growth with $r = 1.2/\text{hr}$ and $K = 10^6$ cells from $P_0 = 100$. Without solving for $t$, find the population of fastest growth and the maximum growth rate (cells/hr). (Recall: fastest at $P = K/2$, rate $rK/4$ — §19.7.)

19.36 ⭐⭐⭐⭐ Technology adoption (economics). A new app's user base follows logistic growth toward $K = 2$ million users with $r = 0.4/\text{month}$, starting at $P_0 = 5{,}000$. Find the month in which adoption is fastest (the inflection point), and the user count there. Then explain, in terms of the S-curve, what a product manager should conclude about whether the steepest growth is ahead or behind once the user base reaches $1.2$ million.

Part I — The SIR Model (§19.9)

19.37 ⭐⭐ For an SIR model with $\beta = 3\times 10^{-4}$, $\gamma = 0.1$, and $N = 1000$, compute the basic reproduction number $R_0 = \beta N / \gamma$. Does the epidemic grow or die out, and why?

19.38 ⭐⭐⭐ For the parameters of 19.37, find the susceptible level $S^\*$ at which the infectious curve $I(t)$ peaks. (Hint: $dI/dt = 0$ when $\beta S - \gamma = 0$, so $S^\* = \gamma/\beta = N/R_0$ — §19.9.) What fraction of the population is still susceptible at the peak?

19.39 ⭐⭐⭐ Herd immunity (public health). Compute the herd immunity threshold $p = 1 - 1/R_0$ for $R_0 = 1.3,\ 3,\ 5,\ 12,\ 18$. Which diseases (influenza, measles, smallpox) do these roughly correspond to, and what vaccination coverage does each demand?

19.40 ⭐⭐⭐⭐ Vaccination compartment (epidemiology). Modify the SIR system so that susceptibles are vaccinated at a constant per-capita rate $v$, moving directly from $S$ to $R$. Write the new system of three ODEs. Explain why adding the $-vS$ term lowers the effective reproduction number, and argue (using the early-outbreak linearization of §19.9) how large $v$ must be to drive effective growth below the $R_0 = 3$ threshold.

Difficulty Distribution

Total: 40 problems (6 + 18 + 12 + 4).

Fields represented (applied problems): nuclear physics (19.25), pharmacology (19.26), finance (19.27), environmental/chemical engineering (19.28–19.30), forensics (19.32), food science (19.33), economics / technology adoption (19.36), public health and epidemiology (19.37–19.40). The SIR work in Part I feeds directly into the Chapter 39 capstone, where you calibrate $\beta$ and $\gamma$ to real data.

Worked solutions to selected problems appear in appendices/answers-to-selected.md.