Applications and initial-value problems

The reason differential equations matter outside the classroom is that nature respects rates. Whenever the rate of change of a quantity depends on the quantity itself , exponential growth, exponential decay, Newtonian cooling, mixing problems , you have a differential equation. This section is about modelling: translating a story into a DE, solving with the appropriate method, and applying the initial condition to pin down the constant.

Pattern 1: Growth and decay

"The rate of change of a quantity is proportional to the quantity itself."

Translation: $\dfrac{dy}{dt} = ky$ where $k > 0$ for growth and $k < 0$ for decay. Separable. Solution: $y(t) = y(0) e^{kt}$.

Examples: population growth (when resources are abundant), radioactive decay (with $k = -\lambda$), bacterial cultures, compound interest. The doubling time for growth is $\ln 2 / k$; the half-life for decay is $\ln 2 / |k|$.

Pattern 2: Newton's law of cooling

"The rate of cooling of a body is proportional to the difference between its temperature and the surroundings' temperature."

Translation: $\dfrac{dT}{dt} = -k(T - T_{\text{env}})$, with $k > 0$ and $T(0) = T_0$.

This is separable: $\dfrac{dT}{T - T_{\text{env}}} = -k\,dt$. Solution: $T(t) = T_{\text{env}} + (T_0 - T_{\text{env}})e^{-kt}$.

As $t \to \infty$, $T \to T_{\text{env}}$, as expected. The constant $k$ is determined by additional data (e.g. the temperature at one specific time).

Pattern 3: Mixing problems

A tank holds $V$ litres of brine with $y(t)$ kilograms of salt at time $t$. Brine of concentration $c_{\text{in}}$ flows in at $r_{\text{in}}$ L/min, and well-mixed brine flows out at $r_{\text{out}}$ L/min.

Rate of salt in $= r_{\text{in}} c_{\text{in}}$. Rate of salt out $= r_{\text{out}} \cdot \dfrac{y(t)}{V(t)}$, where $V(t) = V_0 + (r_{\text{in}} - r_{\text{out}}) t$.

If $r_{\text{in}} = r_{\text{out}}$, volume is constant, and the DE is linear with constant coefficients. If they differ, $V$ is linear in $t$, and the DE is linear with variable coefficients , but still linear, so the integrating-factor method applies.

Pattern 4: RC and RL circuits

A capacitor of capacitance $C$ in series with a resistor $R$ and a voltage source $V(t)$: $RC\dfrac{dq}{dt} + q = CV(t)$. Linear in $q$, integrating-factor method.

A coil of inductance $L$ in series with $R$ and voltage source: $L\dfrac{di}{dt} + R i = V(t)$. Linear in $i$.

Pattern 5: Bacterial doubling and radioactive half-life

If a population doubles in time $T_d$, then $2y_0 = y_0 e^{kT_d}$, so $k = \dfrac{\ln 2}{T_d}$. Similarly half-life: $k_{\text{decay}} = -\dfrac{\ln 2}{T_{1/2}}$.

Worked examples

Example 1. A population of bacteria doubles every 3 hours. Find its size after 10 hours if initially there are $1000$ bacteria.

Model: $\dfrac{dy}{dt} = ky$, so $y(t) = 1000 e^{kt}$. Doubling: $y(3) = 2000 \Rightarrow e^{3k} = 2 \Rightarrow k = \dfrac{\ln 2}{3}$. At $t = 10$: $y(10) = 1000 \cdot e^{(10\ln 2)/3} = 1000 \cdot 2^{10/3} \approx 1000 \cdot 10.08 \approx 10079$.

Example 2. A body cools from $80\,^\circ\text{C}$ to $60\,^\circ\text{C}$ in $10$ minutes in a $20\,^\circ\text{C}$ environment. What temperature does it reach after $20$ minutes?

Newton's cooling: $T(t) = 20 + 60 e^{-kt}$ (since $T_0 - T_{\text{env}} = 60$). At $t = 10$: $60 = 20 + 60 e^{-10k} \Rightarrow 40 = 60 e^{-10k} \Rightarrow e^{-10k} = 2/3$. So $e^{-20k} = (2/3)^2 = 4/9$. Hence $T(20) = 20 + 60 \cdot 4/9 = 20 + \dfrac{80}{3} = \dfrac{60 + 80}{3} \cdot 1 = \dfrac{140}{3} \approx 46.67\,^\circ\text{C}$.

Example 3. A radioactive substance has a half-life of $1620$ years. How long does it take to decay to $10\%$ of its initial mass?

$y(t) = y_0 e^{-\lambda t}$ with $\lambda = \ln 2 / 1620$. Solve $0.1 y_0 = y_0 e^{-\lambda t} \Rightarrow t = \dfrac{\ln 10}{\lambda} = \dfrac{1620 \ln 10}{\ln 2} \approx 1620 \cdot 3.322 \approx 5382$ years.

Example 4. A tank contains $100$ L of brine with $5$ kg salt. Fresh water enters at $4$ L/min, and the mixture leaves at $4$ L/min. Find the amount of salt at time $t$.

Volume constant at $100$ L. Salt in $= 0$. Salt out $= 4 \cdot y/100$. So $\dfrac{dy}{dt} = -\dfrac{y}{25}$. Solution: $y = 5 e^{-t/25}$.

Example 5. A curve passes through $(1, 2)$ and at every point its slope is $\dfrac{y}{x}$. Find the equation of the curve.

DE: $\dfrac{dy}{dx} = \dfrac{y}{x}$. Separable. Solution $y = Cx$. Apply IC: $2 = C \cdot 1$, so $C = 2$. Curve: $y = 2x$.

Example 6. A curve in the first quadrant has the property that the slope at any point $(x, y)$ equals $\dfrac{x + y}{x}$, and passes through $(1, 1)$. Find it.

DE: $\dfrac{dy}{dx} = 1 + \dfrac{y}{x}$. Homogeneous. Substitute $v = y/x$: $v + xv' = 1 + v$, so $xv' = 1$, integrate: $v = \ln x + C$, hence $y = x \ln x + Cx$. Apply IC: $1 = 0 + C$, so $C = 1$. Curve: $y = x\ln x + x = x(\ln x + 1)$.

Try it yourself

1. A bacterial culture doubles in 4 hours. How many will there be in 12 hours if it starts with $500$?
2. A radioactive isotope has half-life 7 days. After 21 days, what fraction remains?
3. Newton's cooling: a cup of coffee at $90\,^\circ\text{C}$ cools to $70\,^\circ\text{C}$ in 5 minutes (room $20\,^\circ\text{C}$). Find $T$ at $t = 15$ min.
4. A tank of $200$ L brine, initially salt-free, has brine of concentration $0.5$ kg/L flowing in at $4$ L/min and mixture out at $4$ L/min. Find $y(t)$.
5. Find the curve passing through $(0, 1)$ with $\dfrac{dy}{dx} = y \cdot \cos x$.
6. A curve in the first quadrant passes through $(1, 0)$ and has slope $\dfrac{y - x}{y + x}$. Find it.
7. Money grows at $5\%$ per year compounded continuously. Time to triple?
8. A snowball melts at a rate proportional to its surface area. If $V$ is volume and $S =$ surface area, model $V(t)$. (Use $V = \frac{4}{3}\pi r^3$, $S = 4\pi r^2$, so $\dfrac{dV}{dt} = -kS$ leads to a DE for $r$.)
9. A tank of $50$ L contains pure water. Brine of $0.2$ kg/L flows in at $2$ L/min; mixture out at $3$ L/min. Find $y(t)$ until the tank is empty.
10. A curve has the property that the tangent at any point passes through $(0, 0)$. What is the curve? (Hint: form a DE.)
11. The half-life of carbon-14 is $5730$ years. A bone has $25\%$ of its original carbon-14. Estimate its age.
12. An RL circuit: $L\dfrac{di}{dt} + Ri = V_0$ (constant). Solve $i(t)$ with $i(0) = 0$.
13. The slope of a curve at $(x, y)$ equals $y/x$ minus $y \cot x$. Find the family of curves.
14. A population of fish in a lake follows $\dfrac{dP}{dt} = 0.1 P - 10$ (logistic-like, but linear). Find $P(t)$ if $P(0) = 200$.

Pitfalls and tricks

• Translate carefully. Read the problem, identify the rate and what it's proportional to, and write the DE before solving.
• Pick the right method. Most growth/decay/cooling problems are separable. Mixing problems with non-constant volume often need integrating factor.
• Apply IC last. Solve the general form, then substitute initial values.
• Check units. Make sure rates and times are in consistent units.
• Sanity check the answer. As $t \to \infty$, does the solution behave physically? Growth diverges, cooling approaches ambient, mixing approaches the inflow concentration. If your answer says otherwise, suspect an error.