# Introduction to Differential Equations

Mathematics I · Integral Calculus · https://tryals.app/learn/mathematics-i/introduction-to-differential-equations

## The Language of Change

A **differential equation** relates an unknown function to its **derivatives**, describing how quantities evolve in the physical world. For example, $\frac{dy}{dx} = 2y$ states that the rate of change of $y$ equals twice $y$.

A **solution** is a function that satisfies the equation. The **order** of a differential equation is the order of the highest derivative appearing in it.

Differential equations typically have a **family** of solutions containing a constant of integration. An **initial condition** (like $y(0) = 3$) pins down one particular solution.

| Concept | Meaning | Example |
| --- | --- | --- |
| **General Solution** | Solution with an unknown constant $C$ | $y = Ce^{kx}$ |
| **Particular Solution** | Constant resolved using an initial point | $y = 3e^{kx}$ |

## Separable Equations

A **first-order separable** equation can be manipulated so that each variable is on its own side:

$$\frac{dy}{dx} = g(x)h(y) \;\Rightarrow\; \frac{dy}{h(y)} = g(x)\, dx$$

To solve, **integrate both sides**. For growth $\frac{dy}{dx} = ky$: separate to $\frac{dy}{y} = k\, dx$, integrate to $\ln|y| = kx + C$, and exponentiate to the exponential growth law $y = y_0 e^{kx}$.

This method solves radioactive decay, Newton's law of cooling $\frac{dT}{dt} = -k(T - T_{env})$, and population models.

> **Common Pitfall:** Forgetting the **constant of integration** when integrating both sides. Omitting $+C$ treats the general solution as a single function and loses the infinite family of solutions.

## Practice questions

5 of this lesson's 10 practice questions, with answers. The full set is in the app.

### 1. Order the steps for solving a first-order separable differential equation with an initial condition.

**Answer:**

1. Write the equation in the form dy/dx = g(x) h(y)
2. Separate the variables so that all y-terms sit with dy and all x-terms with dx
3. Integrate both sides, introducing a single constant of integration
4. Rearrange to express y explicitly, giving the general solution
5. Substitute the initial condition and solve for the constant

**Why:** Separate, integrate, solve, then apply the condition. Using the initial condition too early, or omitting + C when integrating, both destroy the method: without the constant there is nothing for the condition to fix.

Page: https://tryals.app/practice/mathematics-i/introduction-to-differential-equations/order-the-steps-for-solving-a-first-order-separable-differential

### 2. The order of a differential equation determines how many independent constants its general solution carries.

**Answer:** True

**Why:** True. Solving an $n$th-order equation takes $n$ integrations, each contributing a constant, so the general solution has $n$ of them, and $n$ initial conditions are needed to pin them down. That is why a second-order equation of motion needs both an initial position and an initial velocity.

Page: https://tryals.app/practice/mathematics-i/introduction-to-differential-equations/the-order-of-a-differential-equation-determines-how-many-independent

### 3. Arc length, volume, work and centre of mass all end up as integrals. What do these quantities share that makes one operation serve them all?

A. That integration was generalised separately to each after being invented for area
B. That they are all proportional to length
C. That they are all geometric measurements of the same curve
D. Additivity over pieces, and near-simplicity on a small piece

**Answer:** D. Additivity over pieces, and near-simplicity on a small piece

**Why:** Two conditions, neither of them geometric: the total is the sum over pieces, and each small piece is approximately something elementary. Whenever both hold you can write an element — dL, dW, dV — and the integral assembles it. Setting up the element is the transferable skill; the formulas are downstream.

Page: https://tryals.app/practice/mathematics-i/introduction-to-differential-equations/arc-length-volume-work-and-centre-of-mass-all-end-up-as-integrals

### 4. The average value of f on [a,b] is (1/(b−a))∫f. Why is that the right definition, when an average is normally a sum divided by a count and there are infinitely many values here?

A. Because averages of infinitely many values are undefined, so this is a convention adopted by analogy.
B. Because the mean value theorem for integrals defines the average, so no further justification is available.
C. Because the integral already computes an average, and dividing by the interval length converts it to a total.
D. Because the integral plays the sum's role and the length the count's.

**Answer:** D. Because the integral plays the sum's role and the length the count's.

**Why:** Sample at n points and average; multiply and divide by the spacing and it becomes (1/(b−a))·Σf(xᵢ)Δx, a Riemann sum. So it is not an analogy, the integral does the summing and (b−a) does the counting. The Mean Value Theorem for Integrals adds that a continuous function actually attains its average somewhere, which is the derivative statement on the integral side.

Page: https://tryals.app/practice/mathematics-i/introduction-to-differential-equations/the-average-value-of-f-on-a-b-is-1-b-a-f-why-is-that-the-right

### 5. The "split into thin pieces" argument gives the right arc length with $ds$ but the wrong surface area if you use $dx$ instead. What fails?

A. Nothing fails; both give the same answer
B. The piece is no longer nearly simple where it matters
C. The limit fails to exist for surfaces of revolution
D. The pieces stop being additive once a surface is involved

**Answer:** B. The piece is no longer nearly simple where it matters

**Why:** The staircase paradox: the shape converges and the length does not. Additivity is not enough, the piece must be nearly simple in the quantity being measured, which is why $ds$ and not $dx$ appears.

Page: https://tryals.app/practice/mathematics-i/introduction-to-differential-equations/the-split-into-thin-pieces-argument-gives-the-right-arc-length-with