1. Basic Concepts

Differential equations are equations that relate a function to its derivatives. They appear throughout science and engineering, modeling everything from population growth to planetary motion. We begin by distinguishing between two major types.

Definition 1.1: Ordinary Differential Equation (ODE)

An ordinary differential equation is an equation involving an unknown function $x(t)$ of a single independent variable $t$ and one or more of its derivatives. The general form is:

F\left(t, x, \frac{dx}{dt}, \frac{d^2x}{dt^2}, \ldots, \frac{d^nx}{dt^n}\right) = 0

In contrast, a partial differential equation (PDE) involves partial derivatives with respect to multiple independent variables.

Example 1.1: ODE vs PDE

ODEs:

$\frac{dx}{dt} = kx$ (exponential growth/decay)
$\frac{d^2x}{dt^2} + \omega^2 x = 0$ (simple harmonic motion)
$\frac{dx}{dt} = rx(1-x/K)$ (logistic equation)

PDEs:

$\frac{\partial u}{\partial t} = k\frac{\partial^2 u}{\partial x^2}$ (heat equation)
$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$ (Laplace's equation)

Definition 1.2: Order of a Differential Equation

The order of a differential equation is the highest order derivative that appears in the equation. A first-order ODE involves only $\frac{dx}{dt}$ , a second-order ODE involves $\frac{d^2x}{dt^2}$ , and so on.

Definition 1.3: Linear and Nonlinear ODEs

An $n$ th-order ODE is linear if it can be written in the form:

a_n(t)\frac{d^nx}{dt^n} + a_{n-1}(t)\frac{d^{n-1}x}{dt^{n-1}} + \cdots + a_1(t)\frac{dx}{dt} + a_0(t)x = f(t)

where the coefficients $a_i(t)$ and forcing term $f(t)$ depend only on $t$ , not on $x$ or its derivatives. Otherwise, the ODE is nonlinear.

Example 1.2: Linear vs Nonlinear

Linear ODEs:

$x'' + 3x' + 2x = \sin t$ (second-order linear)
$x' + p(t)x = q(t)$ (first-order linear)

Nonlinear ODEs:

$x' = x^2$ (nonlinear in $x$ )
$x'' + \sin x = 0$ (nonlinear due to $\sin x$ )
$x' \cdot x'' = t$ (nonlinear in derivatives)

Definition 1.4: Solution Types

A solution to an ODE on an interval $I$ is a function $x = \phi(t)$ that satisfies the equation for all $t \in I$ .

General solution: Contains arbitrary constants (as many as the order of the ODE)
Particular solution: A specific solution obtained by assigning values to the constants
Singular solution: A solution that cannot be obtained from the general solution

Definition 1.5: Initial Value Problem (IVP)

An initial value problem (or Cauchy problem) consists of a differential equation together with initial conditions that specify the value of the solution and its derivatives at a specific point:

\begin{cases} \frac{dx}{dt} = f(t, x) \\ x(t_0) = x_0 \end{cases}

For higher-order equations, additional initial conditions are needed for the derivatives.

2. Direction Fields and Geometric Interpretation

Before solving ODEs analytically, we can gain valuable insight through geometric visualization. For a first-order equation $\frac{dx}{dt} = f(t, x)$ , the function $f(t, x)$ gives the slope of the solution curve at each point $(t, x)$ .

Definition 2.1: Direction Field (Slope Field)

A direction field for the equation $\frac{dx}{dt} = f(t, x)$ is a collection of short line segments drawn at points $(t, x)$ in the plane, where each segment has slope $f(t, x)$ . Solution curves are tangent to these segments at every point.

Definition 2.2: Isocline

An isocline is a curve in the $(t, x)$ -plane along which the slope $f(t, x)$ is constant. For a slope value $m$ , the isocline is the set of points satisfying:

f(t, x) = m

The nullcline (zero isocline) is where $f(t, x) = 0$ , corresponding to horizontal tangent lines. Points on the nullcline may be equilibrium solutions.

Example 2.1: Direction Field Analysis

Consider the logistic equation $\frac{dx}{dt} = x(1-x)$ .

Nullcline analysis: Setting $x(1-x) = 0$ gives $x = 0$ and $x = 1$ .

For $0 < x < 1$ : $x(1-x) > 0$ , so slopes are positive (solutions increase)
For $x < 0$ or $x > 1$ : $x(1-x) < 0$ , so slopes are negative (solutions decrease)

This tells us that $x = 0$ is an unstable equilibrium (solutions move away) and $x = 1$ is a stable equilibrium (solutions approach it).

Remark:

Direction fields are especially useful when analytical solutions are difficult or impossible to find. They provide qualitative information about solution behavior: where solutions increase/decrease, equilibria, and asymptotic behavior.

3. Separable Equations

The simplest first-order ODEs to solve are separable equations, where the variables can be completely separated to opposite sides of the equation.

Definition 3.1: Separable Equation

A first-order ODE is separable if it can be written in the form:

\frac{dx}{dt} = f(t) \cdot g(x)

where the right-hand side is a product of a function of $t$ alone and a function of $x$ alone.

Algorithm 3.1: Solving Separable Equations

Input: Separable equation $\frac{dx}{dt} = f(t)g(x)$

Output: General solution $x(t)$

1. Separate variables: Write $\frac{dx}{g(x)} = f(t)\,dt$

2. Integrate both sides: $\int \frac{1}{g(x)}\,dx = \int f(t)\,dt + C$

3. Solve for x: If possible, express $x$ explicitly as a function of $t$

4. Check for singular solutions: Points where $g(x) = 0$ may give additional solutions

Example 3.1: Basic Separable Equation

Solve: $\frac{dx}{dt} = tx$

Solution:

Step 1: Separate variables (assuming $x \neq 0$ ):

\frac{dx}{x} = t\,dt

Step 2: Integrate both sides:

\int \frac{dx}{x} = \int t\,dt \implies \ln|x| = \frac{t^2}{2} + C

Step 3: Solve for $x$ :

|x| = e^{t^2/2 + C} = e^C \cdot e^{t^2/2} \implies x = Ae^{t^2/2}

where $A = \pm e^C$ is an arbitrary constant. Note that $x = 0$ (when $A = 0$ ) is also a solution.

Example 3.2: Separable Equation with IVP

Solve the IVP: $\frac{dx}{dt} = \frac{t}{x}$ , $x(0) = 2$

Solution:

Separating: $x\,dx = t\,dt$

Integrating:

\frac{x^2}{2} = \frac{t^2}{2} + C \implies x^2 = t^2 + 2C

Using initial condition $x(0) = 2$ :

4 = 0 + 2C \implies C = 2

Therefore:

x^2 = t^2 + 4 \implies x = \sqrt{t^2 + 4}

(taking positive root since $x(0) = 2 > 0$ )

Theorem 3.1: Existence of Solutions for Separable Equations

For the separable equation $\frac{dx}{dt} = f(t)g(x)$ , if $f$ is continuous on an interval containing $t_0$ and $g$ is continuous on an interval containing $x_0$ with $g(x_0) \neq 0$ , then the IVP with $x(t_0) = x_0$ has a unique solution in some neighborhood of $t_0$ .

Example 3.3: Singular Solutions

Solve: $\frac{dx}{dt} = x^{2/3}$

Solution:

For $x \neq 0$ , separating and integrating:

\int x^{-2/3}\,dx = \int dt \implies 3x^{1/3} = t + C

x = \left(\frac{t + C}{3}\right)^3

However, $x = 0$ is also a solution (singular solution).

Important: The IVP with $x(0) = 0$ has infinitely many solutions! This shows uniqueness can fail when $g(x_0) = 0$ .

4. Homogeneous Equations

Another important class of first-order ODEs are homogeneous equations, which can be transformed into separable equations through an appropriate substitution.

Definition 4.1: Homogeneous Function

A function $F(t, x)$ is homogeneous of degree $n$ if for all $\lambda > 0$ :

F(\lambda t, \lambda x) = \lambda^n F(t, x)

Example 4.1: Homogeneous Functions

$F(t,x) = t^2 + x^2$ is homogeneous of degree 2: $(\lambda t)^2 + (\lambda x)^2 = \lambda^2(t^2 + x^2)$
$F(t,x) = \frac{x}{t}$ is homogeneous of degree 0: $\frac{\lambda x}{\lambda t} = \frac{x}{t}$
$F(t,x) = t + x^2$ is not homogeneous

Definition 4.2: Homogeneous First-Order ODE

A first-order ODE $\frac{dx}{dt} = F(t, x)$ is homogeneous if $F(t, x)$ is homogeneous of degree 0. Equivalently, the equation can be written as:

\frac{dx}{dt} = G\left(\frac{x}{t}\right)

for some function $G$ .

Algorithm 4.1: Solving Homogeneous Equations

Input: Homogeneous equation $\frac{dx}{dt} = G(x/t)$

Output: General solution $x(t)$

1. Substitute: Let $u = x/t$ , so $x = ut$

2. Differentiate: $\frac{dx}{dt} = u + t\frac{du}{dt}$

3. Transform equation: $u + t\frac{du}{dt} = G(u)$

4. Separate: $\frac{du}{G(u) - u} = \frac{dt}{t}$

5. Integrate both sides

6. Back-substitute: Replace $u$ with $x/t$

Example 4.2: Solving a Homogeneous Equation

Solve: $\frac{dx}{dt} = \frac{t + x}{t}$

Solution:

First verify it's homogeneous: $\frac{t+x}{t} = 1 + \frac{x}{t} = G(x/t)$ ✓

Let $u = x/t$ , so $x = ut$ and $\frac{dx}{dt} = u + t\frac{du}{dt}$ .

Substituting:

u + t\frac{du}{dt} = 1 + u \implies t\frac{du}{dt} = 1

This is now separable:

du = \frac{dt}{t} \implies u = \ln|t| + C

Back-substituting $u = x/t$ :

\frac{x}{t} = \ln|t| + C \implies x = t\ln|t| + Ct

Example 4.3: More Complex Homogeneous Equation

Solve: $\frac{dx}{dt} = \frac{x^2 + t^2}{2tx}$

Solution:

Check homogeneity: $\frac{x^2 + t^2}{2tx} = \frac{(x/t)^2 + 1}{2(x/t)} = \frac{u^2 + 1}{2u}$ where $u = x/t$ . ✓

With $x = ut$ :

u + t\frac{du}{dt} = \frac{u^2 + 1}{2u}

t\frac{du}{dt} = \frac{u^2 + 1}{2u} - u = \frac{u^2 + 1 - 2u^2}{2u} = \frac{1 - u^2}{2u}

Separating:

\frac{2u\,du}{1 - u^2} = \frac{dt}{t}

Integrating (LHS with substitution $v = 1 - u^2$ ):

-\ln|1 - u^2| = \ln|t| + C_1 \implies |1 - u^2| = \frac{C}{|t|}

Back-substituting and simplifying:

1 - \frac{x^2}{t^2} = \frac{C}{t} \implies t^2 - x^2 = Ct

Remark:

Homogeneous equations of the form $M(t,x)\,dt + N(t,x)\,dx = 0$ where both $M$ and $N$ are homogeneous of the same degree can be solved similarly. The substitution $x = ut$ (or $t = vx$ ) always works.

5. Applications: Population Models

Differential equations are powerful tools for modeling real-world phenomena. We examine several population models that lead to separable equations.

5.1 Exponential Growth/Decay (Malthusian Model)

The simplest population model assumes the rate of change is proportional to the current population:

\frac{dN}{dt} = rN

where $r$ is the intrinsic growth rate. This separable equation has solution:

N(t) = N_0 e^{rt}

For $r > 0$ : exponential growth. For $r < 0$ : exponential decay.

5.2 Logistic Growth

The logistic model accounts for limited resources by introducing a carrying capacity $K$ :

\frac{dN}{dt} = rN\left(1 - \frac{N}{K}\right)

Example 5.1: Solving the Logistic Equation

This is a separable equation. Using partial fractions:

\frac{dN}{N(1 - N/K)} = r\,dt

\frac{1}{N(1-N/K)} = \frac{1}{N} + \frac{1/K}{1-N/K} = \frac{1}{N} + \frac{1}{K-N}

Integrating:

\ln|N| - \ln|K-N| = rt + C

\ln\left|\frac{N}{K-N}\right| = rt + C

Solving for $N$ with initial condition $N(0) = N_0$ :

N(t) = \frac{K}{1 + \left(\frac{K}{N_0} - 1\right)e^{-rt}}

Key properties:

$N(t) \to K$ as $t \to \infty$
S-shaped (sigmoid) growth curve
Maximum growth rate at $N = K/2$

5.3 Epidemic Models

Definition 5.1: SI Model

In the SI (Susceptible-Infected) model, let $i(t)$ be the fraction of infected individuals. The model assumes that infected individuals meet susceptibles at a rate proportional to both populations:

\frac{di}{dt} = \lambda i(1-i)

where $\\lambda$ is the contact rate. This is mathematically identical to the logistic equation with $K = 1$ .

Example 5.2: SI Model Solution

With initial infection fraction $i_0$ , the solution is:

i(t) = \frac{1}{1 + \left(\frac{1}{i_0} - 1\right)e^{-\lambda t}}

For example, if $i_0 = 0.01$ and $\\lambda = 0.5$ , then $i(t) \to 1$ as $t \to \infty$ , meaning eventually everyone gets infected.

Definition 5.2: SIS Model

In the SIS model, infected individuals can recover and become susceptible again:

\frac{di}{dt} = \lambda i(1-i) - \gamma i = i(\lambda(1-i) - \gamma)

where $\\gamma$ is the recovery rate.

Theorem 5.1: SIS Epidemic Threshold

For the SIS model, define the basic reproduction number $R_0 = \lambda/\gamma$ .

If $R_0 < 1$ : The infection dies out ( $i(t) \to 0$ )
If $R_0 > 1$ : The infection persists at endemic level $i^* = 1 - 1/R_0$

Note:

The basic reproduction number $R_0$ represents the average number of secondary infections caused by one infected individual in a fully susceptible population. It is a key concept in epidemiology.

Introduction and Elementary Methods

1. Basic Concepts

Definition 1.1: Ordinary Differential Equation (ODE)

Example 1.1: ODE vs PDE

Definition 1.2: Order of a Differential Equation

Definition 1.3: Linear and Nonlinear ODEs

Example 1.2: Linear vs Nonlinear

Definition 1.4: Solution Types

Definition 1.5: Initial Value Problem (IVP)

2. Direction Fields and Geometric Interpretation

Definition 2.1: Direction Field (Slope Field)

Definition 2.2: Isocline

Example 2.1: Direction Field Analysis

Remark:

3. Separable Equations

Definition 3.1: Separable Equation

Algorithm 3.1: Solving Separable Equations

Example 3.1: Basic Separable Equation

Example 3.2: Separable Equation with IVP

Theorem 3.1: Existence of Solutions for Separable Equations

Example 3.3: Singular Solutions

4. Homogeneous Equations

Definition 4.1: Homogeneous Function

Example 4.1: Homogeneous Functions

Definition 4.2: Homogeneous First-Order ODE

Algorithm 4.1: Solving Homogeneous Equations

Example 4.2: Solving a Homogeneous Equation

Example 4.3: More Complex Homogeneous Equation

Remark:

5. Applications: Population Models

5.1 Exponential Growth/Decay (Malthusian Model)

5.2 Logistic Growth

Example 5.1: Solving the Logistic Equation

5.3 Epidemic Models

Definition 5.1: SI Model

Example 5.2: SI Model Solution

Definition 5.2: SIS Model

Theorem 5.1: SIS Epidemic Threshold

Note:

Frequently Asked Questions

What is the difference between an ODE and a PDE?

How do I know if an equation is separable?

What is a direction field used for?

What happens when the separation method fails?

Why do we use the substitution u = x/t for homogeneous equations?

What is the biological meaning of the logistic equation?