University Differential Equations — Modelling with First-Order Equations Worksheet

All 9 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one.

1. Q1Translating a Rate Statement into a Differential Equation

   For each statement, name the unknown function and its independent variable, write the differential equation it describes (use $k$ for the constant of proportionality and say what sign it has), and say whether the equation is linear or nonlinear.

   1. On a frozen pond, the thickness $h$ of the ice, in centimetres, increases with time $t$, in hours, at a rate inversely proportional to the thickness already present.
   2. In a town of $8000$ residents, the number who have heard a rumour increases with time $t$, in days, at a rate jointly proportional to the number who have heard it and the number who have not.
   3. Sunlight entering murky water loses intensity with depth: the intensity $I$ decreases with the depth $x$ below the surface, in metres, at a rate proportional to the intensity at that depth.
2. Q2Translating a Rate Statement into a Differential Equation

   Dead leaves fall on a patch of forest floor at a constant rate of $4\text{kg}/{\text{m}}^2$ per year, and the litter on the ground decomposes at a rate proportional to the amount present, with constant of proportionality $0.5$ per year. The patch is cleared completely at $t=0$. Let $L(t)$ be the litter on the ground, in $\text{kg}/{\text{m}}^2$, $t$ years later.

   1. Write the initial-value problem for $L$.
   2. Give the units of ${\displaystyle \frac{dL}{dt}}$ and of each term on the right-hand side, and confirm that they agree.
   3. Without solving the equation, find the amount of litter the patch approaches in the long run, and explain why that value is the one to expect.
3. Q3Newton's Law of Cooling and Heating

   A steel bolt at $840{}^{\circ }\text{C}$ is plunged into a large oil bath held at $40{}^{\circ }\text{C}$. Newton's law of cooling says that the rate of change of the bolt's temperature is proportional to the difference between the bath temperature and the bolt's temperature. Two minutes later the bolt is at $240{}^{\circ }\text{C}$. Let $T(t)$ be its temperature, in ${}^{\circ }\text{C}$, $t$ minutes after it enters the bath.

   1. Write the initial-value problem and solve it, finding the constant exactly.
   2. When does the bolt reach $90{}^{\circ }\text{C}$?
4. Q4Mixing Problems with Unequal Flow Rates

   A hydroponic reservoir can hold $100\text{L}$. It starts with $50\text{L}$ of water in which $10\text{g}$ of nutrient is dissolved. A solution containing $1\text{g}/\text{L}$ of nutrient flows in at $3\text{L}/\text{min}$, the reservoir is kept well stirred, and the mixture is pumped out at $2\text{L}/\text{min}$. Let $x(t)$ be the mass of nutrient in the reservoir, in grams, after $t$ minutes.

   1. Find the volume $V(t)$ and write the initial-value problem for $x$.
   2. Solve it.
   3. How much nutrient is in the reservoir at the moment it starts to overflow, and what is the concentration then?
5. Q5Mixing Problems with Unequal Flow Rates

   A $120\text{L}$ tank holds a solution containing $6\text{kg}$ of disinfectant. A solution with $0.1\text{kg}/\text{L}$ of disinfectant flows in at $2\text{L}/\text{min}$, and the well-stirred mixture drains at $5\text{L}/\text{min}$. Let $x(t)$ be the mass of disinfectant in the tank, in kilograms, after $t$ minutes. Do not solve the equation.

   1. Find the volume $V(t)$ and the time at which the tank is empty.
   2. Write the initial-value problem for $x$, and state the interval of $t$ on which the model applies.
   3. Is the equation separable? Is it linear? Justify both answers.
6. Q6First-Order Electric Circuits

   A series circuit has a resistor of $10\Omega$, an inductor of $2\text{H}$ and a $40\text{V}$ battery switched in at $t=0$, when no current flows. Kirchhoff's voltage law gives $L{\displaystyle \frac{di}{dt}}+Ri=E$, where $i(t)$ is the current in amperes and $t$ is in seconds.

   1. Solve for $i(t)$ by an integrating factor.
   2. State the steady-state current and the time at which the current reaches $3\text{A}$.
7. Q7First-Order Electric Circuits

   A series circuit has a resistor of $20\Omega$, a capacitor of $0.01\text{F}$ and a $50\text{V}$ source. When the switch closes at $t=0$ the capacitor already carries a charge of $0.1\text{C}$. The charge $q(t)$, in coulombs, satisfies $R{\displaystyle \frac{dq}{dt}}+{\displaystyle \frac{q}{C}}=E$, and the current is $i={\displaystyle \frac{dq}{dt}}$; $t$ is in seconds.

   1. Find $q(t)$ and $i(t)$.
   2. When does the charge reach $0.3\text{C}$?
8. Q8The Logistic Equation

   A bird population on an island, $P(t)$ birds after $t$ years, follows the logistic equation $$\frac{dP}{dt}=0.5P(1-\frac{P}{1200}),P(0)=200.$$

   1. Solve for $P(t)$.
   2. When is the population growing fastest, and at what rate is it growing then?
9. Q9Synthesis — drawing on several topics in this unit

   An insulated $100\text{L}$ hot-water tank is full of water at $70{}^{\circ }\text{C}$. From $t=0$, cold water at $10{}^{\circ }\text{C}$ flows in at $5\text{L}/\text{min}$, the tank is kept well mixed, and water leaves at $5\text{L}/\text{min}$. Treat the heat content of the tank as proportional to volume times temperature, and assume the rate of change of the heat content is the heat carried in minus the heat carried out; $t$ is in minutes.

   1. Show that the tank temperature $T(t)$ satisfies ${\displaystyle \frac{dT}{dt}}={\displaystyle \frac{1}{20}}(10-T)$, and point out which law of heat transfer the equation has the same form as.
   2. Solve it, and find when the water leaving the tank has cooled to $40{}^{\circ }\text{C}$.

The 6 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

From a sentence to an equation

Every model in this set starts with a sentence about a rate. Q1 practises the translation alone: name the unknown function and what it depends on, then turn the sentence into an equation word by word.

The sign of k comes from the sentence, not from the algebra: a quantity that "increases" has a positive rate, one that "decreases" or "loses" a negative one. Decide whether to write k with a minus sign in front or to say k < 0, and say which. In Q1(b), "the number who have not heard" has to be written in terms of the unknown and the town's population. Linearity is then the test from the first set.

Q2 builds a model with two competing effects — a constant input and a loss proportional to the amount present — and adds the initial condition from "cleared completely".

Q2(c) asks for the long-run amount without solving. The amount stops changing when the rate is zero, so set the right-hand side to zero — the equilibrium of the previous set — and explain in words why the two effects balance there.

Newton's law of cooling

The rate of change of an object's temperature is proportional to the difference between the surroundings' temperature and its own. Q3 states the law in words; writing it carefully fixes the sign.

The equation is both linear and separable, and either method works. Substituting u = T − Tₛ reduces it to exponential decay, which is the fastest route. The two data points — the start and the reading two minutes later — fix the constant C and then k; Q3(a) asks for k exactly, so keep it as a logarithm. Q3(b) is solving the solution for t.

Mixing with unequal flow rates

A mixing tank gains solute from the inflow and loses it in the outflow, whose concentration is the tank's own: amount divided by current volume. When the two flow rates differ, the volume changes with time, and the time-dependent volume ends up in a denominator of the equation.

Q4 runs the whole method. Part (c) needs the time of overflow first: set V(t) equal to the capacity. Then evaluate the solution there, and divide by the volume for the concentration.

Q5 stops before solving. The tank drains faster than it fills, so part (a) is when V reaches zero, and part (b) asks for the interval on which the model makes sense — a model that divides by the volume cannot run past the moment the volume is zero. Part (c) asks you to decide separability and linearity and justify both; use the tests from First-Order Solution Methods on the equation you wrote.

First-order circuits

Kirchhoff's voltage law says the voltage drops around a loop add up to the source voltage. For a resistor and an inductor, the unknown is the current; for a resistor and a capacitor, the unknown is the charge, and the current is its derivative. Q6 and Q7 give the two equations.

Both are linear with constant coefficients. Divide by the leading coefficient, use the integrating factor (Q6(a) asks for it by name), and apply the initial condition — zero current in Q6, a given starting charge in Q7. In Q6(b) the steady state is what the current approaches as t grows, which is the term that does not decay. In Q7(a), differentiate q(t) to get i(t).

The logistic equation

dP/dt = rP(1 − P/K) grows almost exponentially while P is small and levels off as P nears the carrying capacity K. It is separable: separate, split the P side by partial fractions, integrate, and solve the logarithms for P. Q8(a) asks for that explicit solution.

Q8(b) asks when growth is fastest. The growth rate dP/dt is a function of P — a quadratic — so maximise it as a function of P first, then use the solution from (a) to find the time at which P reaches that value, and substitute back for the rate.

The synthesis question

Q9 is a mixing problem where the quantity carried by the flow is heat rather than a solute. Part (a) asks you to derive the given equation: the flows in and out are equal, so the volume is constant; write the heat balance with heat proportional to volume times temperature, divide through by the constant of proportionality and the volume, and simplify. Then compare its form with the models earlier in the set. Part (b) is Q3 again — solve and then solve for the time.

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Getting the most out of it

Write the balance in words first

"Rate of change equals rate in minus rate out" is the skeleton of nearly every model here. Write it in words above the equation, then fill in each term.

Carry the units

Check every term of the equation for the same units, and the answer too. It catches a concentration used as a mass and a time constant written upside down.

Ask what happens in the long run

Setting the rate to zero gives the equilibrium; it should match the limit of your solution as t grows. If the two disagree, one of them is wrong.

State where the model applies

A tank that overflows or empties, a population that cannot be negative: the equation outlives the situation it describes. Say when the model stops.

Common questions