Modelling with Differential Equations: Quick Revision

A brief note on Modelling with Differential Equations. Read it before a test. Try each question at the end before you open its answer.

Growth and decay

• Model: a rate proportional to the amount gives $\dfrac{dN}{dt} = kN$ and $N = N_0 e^{kt}$. A negative $k$ is decay.
• Doubling time and half-life: $T = \dfrac{\ln 2}{|k|}$. They do not depend on $N_0$.
• Percentage rates: a yearly rate $r$ gives $k = \ln(1 + r)$. Continuous interest at rate $r$ gives $k = r$.
• Age by decay: $t = \dfrac{T_{1/2}}{\ln 2}\ln\dfrac{N_0}{N}$.
• Logistic: $\dfrac{dP}{dt} = kP\left(1 - \dfrac{P}{M}\right)$ gives $P = \dfrac{M}{1 + Ae^{-kt}}$ with $A = \dfrac{M - P_0}{P_0}$. The growth is fastest at $P = \dfrac{M}{2}$, where it is $\dfrac{kM}{4}$.

Cooling and mixing

• Newton's law: $\dfrac{dT}{dt} = -k(T - T_s)$ gives $T = T_s + (T_0 - T_s)e^{-kt}$. Two readings give $k$.
• Mixing: $\dfrac{dQ}{dt} = r_{\text{in}}c_{\text{in}} - r_{\text{out}}\dfrac{Q}{V}$, with $V = V_0 + (r_{\text{in}} - r_{\text{out}})t$.
• Constant volume: $Q = cV + (Q_0 - cV)e^{-rt/V}$. The steady state is $cV$.
• Euler's method: new value = old value + $h \times$ slope. A smaller step gives a smaller error.

Orthogonal trajectories and curves

• Method: remove the constant to get $\dfrac{dy}{dx} = f(x, y)$. Solve $\dfrac{dy}{dx} = -\dfrac{1}{f(x, y)}$.
• Tangent intercepts: $x - \dfrac{y}{y'}$ on the $x$-axis and $y - xy'$ on the $y$-axis.
• Sub-tangent and sub-normal: $\left|\dfrac{y}{y'}\right|$ and $|yy'|$.

A worked example

A sample decays with half-life $6\ \text{h}$. Find the time for $800$ units to fall to $100$ units.

1. Count the factor. $800 \to 100$ is a factor of $8 = 2^3$, so there are $3$ half-lives.
2. Multiply. The time is $3 \times 6\ \text{h} = 18\ \text{h}$.

Practice with new numbers

A program to try

The program checks that $N = 800e^{-kt}$ with $k = \dfrac{\ln 2}{6}$ halves every $6$ hours.

Mistakes to avoid

• Using $T$ instead of the gap $T - T_s$ in Newton's law.
• Dividing by the starting volume when the volume changes.
• Swapping the slope before removing the constant in orthogonal trajectories.
• Using $k = r$ for a yearly rate. The correct value is $k = \ln(1 + r)$.

Check yourself

1. A population triples in $10$ years under $\dfrac{dN}{dt} = kN$. Find $k$ per year to four decimal places.

2. A cup at $80\ ^\circ\text{C}$ in a room at $20\ ^\circ\text{C}$ is at $50\ ^\circ\text{C}$ after $10$ minutes. What is its temperature after $20$ minutes?

3. A tank of $200\ \text{L}$ has equal inflow and outflow of brine at $0.5\ \text{kg/L}$. Starting from pure water, what amount of salt does it approach?

4. Find the orthogonal trajectories of $y = kx^2$.

5. A curve has a constant sub-normal of $5$. What is its equation, apart from the constant?

Sources

• OpenStax, Calculus Volume 2, Chapter 4, sections 4.2 to 4.5.
• NCERT, Mathematics Part II for Class XII, Chapter 9, the sections on first-order equations and their applications.