# Differential models in one variable

Physics I · Calculus of a Single Variable · https://tryals.app/learn/physics-i/differential-models-in-one-variable

## Differential Models & Separation

A **differential equation** (DE) relates an unknown function to its derivatives, describing how systems evolve in time or space.

**Separable DEs** are equations where algebraic manipulation isolates the variables onto opposite sides:

$$\frac{dy}{dx} = g(x)h(y) \implies \int \frac{1}{h(y)} dy = \int g(x) dx$$

Follow this **Step-by-step method** to solve them:

| Step | Action |
| --- | --- |
| 1. Separate | Move all $y$ terms to $dy$ side, $x$ to $dx$ |
| 2. Integrate | Integrate both sides independently |
| 3. Solve | Isolate $y$ algebraically |
| 4. Condition | Apply initial state to find constant $C$ |

**Initial conditions** supply a known state like $y(t_0)=y_0$ to fix the integration constant $C$ uniquely.

## Classic Models & Pitfalls

**Exponential growth / decay** models systems where change is proportional to current size: $\frac{dN}{dt} = kN$, yielding $N(t) = N_0 e^{kt}$. 

| Parameter | Meaning | Real-World Application |
| --- | --- | --- |
| $k > 0$ | Growth rate | Population, compound interest |
| $k < 0$ | Decay rate | Radioactive decay |

**Newton's law of cooling** states rate of temperature change is proportional to the difference from the environment: $\frac{dT}{dt} = -\lambda(T - T_{\text{env}})$. Its solution is $T(t) = T_{\text{env}} + (T_0 - T_{\text{env}})e^{-\lambda t}$.

**Common pitfall:** Do not confuse rate phrasing. "Grows at a rate proportional to itself" means $\frac{dN}{dt} = kN$ ($e^{kt}$), whereas "grows at a constant rate" means a linear slope. Always check unit consistency for dimensional safety.

## Practice questions

6 of this lesson's 15 practice questions, with answers. The full set is in the app.

### 1. A quantity **doubles every 10 hours**. Without a calculator, estimate how many hours it takes to grow **8-fold**.

**Answer:** 30 (within ±2)

**Why:** Three doublings, 30 hours. Thinking in doublings turns exponentials into counting, which is how epidemiologists and bankers estimate at speed (and why the "rule of 70" works).

Page: https://tryals.app/practice/physics-i/differential-models-in-one-variable/a-quantity-doubles-every-10-hours-without-a-calculator-estimate-how

### 2. Match each differential equation to the story it tells.

**Answer:**

- $y' = ky,\; k > 0$ → Unchecked growth (compound interest)
- $y' = -ky,\; k > 0$ → Decay toward zero (radioactivity)
- $y' = k(A - y)$ → Approach to a ceiling (coffee cooling)
- $y' = ky(1 - y/M)$ → Growth that saturates (logistic population)

**Why:** Four rate laws cover a remarkable share of science. Reading equilibria ($y'=0$) and signs of $y'$ gives you the full qualitative story before any solving, often that is all you need.

Page: https://tryals.app/practice/physics-i/differential-models-in-one-variable/match-each-differential-equation-to-the-story-it-tells

### 3. Complete the description of a separable first-order ODE.

**Answer:** A separable ODE has the form dy/dx = **$g(x) \cdot h(y)$**. We solve it by bringing y-terms to one side: (1/$h(y)$) $dy = g(x)$ dx, then integrating **both** sides independently.

**Why:** Separation works because variables can be moved independently to each side.

Page: https://tryals.app/practice/physics-i/differential-models-in-one-variable/complete-the-description-of-a-separable-first-order-ode

### 4. Money grows as $dP/dt = 0.05P$ (5% continuous interest). A friend says: "It grows by the same amount every year." What is the truth?

A. The friend is right: linear growth of 5% of the initial deposit
B. It grows by the same **percentage** each year
C. It grows more slowly each year as the interest dilutes
D. It alternates between fast and slow years

**Answer:** B. It grows by the same **percentage** each year

**Why:** Proportional rate means exponential growth: $P = P_0e^{0.05t}$. Fixed percentage, compounding absolute gains, the same law behind population booms, radioactive decay (with $k<0$) and epidemic take-offs.

Page: https://tryals.app/practice/physics-i/differential-models-in-one-variable/money-grows-as-dp-dt-0-05p-5-continuous-interest-a-friend-says

### 5. Radioactive decay and Newton's law of cooling produce the same differential equation with different meanings. What do the two systems have in common?

A. That both involve heat transfer at the molecular level
B. That both have solutions only for positive initial conditions
C. That the rate is proportional to the amount or the gap
D. That both approach zero rather than a nonzero equilibrium

**Answer:** C. That the rate is proportional to the amount or the gap

**Why:** A differential equation describes a structure, not a substance. This is why the same mathematics serves nuclear physics, thermodynamics and compound interest, the proportionality is what they have in common.

Page: https://tryals.app/practice/physics-i/differential-models-in-one-variable/radioactive-decay-and-newtons-law-of-cooling-produce-the-same

### 6. Order the steps to solve a linear first-order ODE y' + P(x)$y = Q(x).$

**Answer:**

1. Write in standard form: dy/dx + P(x)$y = Q(x)$
2. Compute integrating factor $\mu$(x) = e^($\int P(x)dx$)
3. Multiply both sides by $\mu$: $d/dx[\mu(x)y] = \mu(x)Q(x)$
4. Integrate both sides and solve for y

**Why:** The integrating factor transforms the left side into an exact derivative, making integration straightforward.

Page: https://tryals.app/practice/physics-i/differential-models-in-one-variable/order-the-steps-to-solve-a-linear-first-order-ode-y-p-x-y-q-x