Introduction to Differential Equations

MATH 341 — Notes 1

University of Scranton

2026-05-31

Goals

What We Will Cover

Define what a differential equation is and what it means for a function to be a solution
Classify ODEs by:
- order
- linearity / nonlinearity
- homogeneity
- constant vs. variable coefficients
- autonomy
Introduce initial value problems (IVPs)
Survey classical sources of ODEs in applications

Note

This material corresponds to Section 1.1 of Logan (2015).

What Is a Differential Equation?

The Big Idea

Key Point

A differential equation is any equation involving an unknown function together with one or more of its derivatives.

The goal is to find a function — not a number.

Contrast with algebra:

Type	Equation	Unknown
Algebraic	\(x^2 - 4 = 0\)	a number \(x = \pm 2\)
Differential	\(\dfrac{dy}{dx} = y\)	a function \(y(x) = Ce^x\)

Important

Solutions to differential equations are functions, not numbers.

ODEs vs. PDEs

Ordinary Differential Equation (ODE): Involves a function of a single independent variable

\[\ddot{x} + \omega^2 x = 0 \quad \leftarrow \text{ one variable } t\]

Partial Differential Equation (PDE): Involves a function of several independent variables

\[u_t = k\,u_{xx} \quad \leftarrow \text{ two variables } x, t \quad \text{(heat equation)}\]

Note

In this course we focus exclusively on ODEs.

Derivative notation is not standardized — see the footnote in the notes for the Leibniz, prime, dot, subscript, and operator conventions.¹

Classifying ODEs

Order

The order of an ODE is the order of the highest derivative present.

Equation	Order
\(y' = ky\)	1st
\(y'' + \omega^2 y = 0\)	2nd
\(y''' - y'' + y = \sin t\)	3rd

General form of an \(n\)th-order ODE: \[F\!\left(t, y, y', y'', \ldots, y^{(n)}\right) = 0\]

or solved for the highest derivative: \[y^{(n)} = f\!\left(t, y, y', \ldots, y^{(n-1)}\right)\]

Linearity

An ODE is linear if it has the form: \[a_n(t)\,y^{(n)} + \cdots + a_1(t)\,y' + a_0(t)\,y = g(t)\]

where the \(a_k(t)\) and \(g(t)\) depend only on \(t\), not on \(y\).

Equation	Linear?	Reason
\(y'' + 3y' - 2y = e^t\)	✓	Coefficients depend only on \(t\)
\(y'' + \sin(y) = 0\)	✗	\(\sin(y)\) is nonlinear in \(y\)
\(y' = y^2\)	✗	\(y^2\) is nonlinear in \(y\)
\(t^2 y'' + ty' + y = 0\)	✓	Coefficients \(t^2, t, 1\) depend only on \(t\)

Homogeneity, Coefficients, Autonomy

Homogeneous — forcing term \(g(t) \equiv 0\): \[a_n(t)\,y^{(n)} + \cdots + a_0(t)\,y = 0\]

Inhomogeneous — \(g(t) \not\equiv 0\).

Tip

The general solution of an inhomogeneous linear ODE = particular solution + general solution of the homogeneous equation.

Constant coefficients — each \(a_k\) is a constant number (not a function of \(t\)).

Variable coefficients — at least one \(a_k\) depends on \(t\).

Autonomous — \(t\) does not appear explicitly on the right-hand side: \[y' = f(y) \;\text{(autonomous)} \qquad \text{vs.} \qquad y' = f(t,y) \;\text{(non-autonomous)}\]

Classification Examples

Example 1 — Logistic Equation

\[\frac{dP}{dt} = rP\!\left(1 - \frac{P}{K}\right)\] 1st-order, nonlinear, autonomous — population growth with carrying capacity \(K\).

Example 2 — Simple Harmonic Oscillator

\[\frac{d^2x}{dt^2} + \omega^2 x = 0\] 2nd-order, linear, homogeneous, constant-coefficient, autonomous

Example 3 — Driven Oscillator

\[\frac{d^2x}{dt^2} + 2\gamma\frac{dx}{dt} + \omega^2 x = F_0\cos(\Omega t)\] 2nd-order, linear, inhomogeneous, non-autonomous, constant coefficients

Classification Examples (cont.)

Example 4 — Euler’s Equation

\[t^2 y'' + t y' - 4y = 0\] 2nd-order, linear, homogeneous, variable (non-constant) coefficients

Solutions to ODEs

What Is a Solution?

A solution of an ODE on an interval \(I\) is a function \(y = \phi(t)\) that:

Is differentiable (sufficiently many times) on \(I\), and
Satisfies the equation for every \(t \in I\).

Example 5 — General and Particular Solutions

Consider \(y' - 2y = 0\). Check \(y(t) = Ce^{2t}\): \[(Ce^{2t})' - 2(Ce^{2t}) = 2Ce^{2t} - 2Ce^{2t} = 0 \checkmark\]

\(y(t) = Ce^{2t}\) for any constant \(C\) — general solution (one-parameter family)
With \(y(0) = 3\): \(C = 3\), so \(y(t) = 3e^{2t}\) — particular solution

A 1st-order ODE \(\to\) one-parameter family of solutions

A 2nd-order ODE \(\to\) two-parameter family, and so on.

Visualizing the Family of Solutions

Figure 1: Family of solutions \(y(t) = Ce^{2t}\) of \(y' - 2y = 0\). Each curve is one member of the family; an initial condition picks exactly one.

Initial Value Problems

Definition

Initial Value Problem (IVP) — 1st Order

\[\begin{cases} y' = f(t, y) \\ y(t_0) = y_0 \end{cases}\]

\(n\)th-order IVP: specify \(y(t_0), y'(t_0), \ldots, y^{(n-1)}(t_0)\).

The initial conditions geometrically select a single solution curve from the entire family.

Existence & Uniqueness — Picard–Lindelöf theorem:

If \(f\) and \(\partial f/\partial y\) are continuous near \((t_0, y_0)\), then the IVP has a unique solution on some interval containing \(t_0\).

IVP Example

Example 6

Solve: \(\quad y' = -3y, \quad y(0) = 5\).

Step 1. General solution: \(y(t) = Ce^{-3t}\).

Step 2. Apply IC: \(C e^0 = 5 \;\Rightarrow\; C = 5\).

Solution: \(\boxed{y(t) = 5e^{-3t}}\)

Figure 2: The initial condition \((0,5)\) selects one curve from the gray family.

Where Do ODEs Come From?

Three Major Sources

ODEs arise wherever a quantity’s rate of change is related to its current state.

We survey three classical sources:

🔧 Classical Mechanics — Newton, Lagrange, Hamilton
⚡ Electrical Circuits — Kirchhoff’s laws
🌱 Population Dynamics — Malthus, Verhulst, Lotka–Volterra

Classical Mechanics

Newton’s Second Law

\[F = ma = m\ddot{x}\]

Spring-mass system (Hooke’s law: \(F = -kx\)):

\[m\ddot{x} = -kx \quad \Longrightarrow \quad \ddot{x} + \omega^2 x = 0, \quad \omega = \sqrt{k/m}\]

2nd-order, linear, homogeneous, constant-coefficient, autonomous
General solution: \(x(t) = A\cos(\omega t) + B\sin(\omega t)\)
Describes pure oscillation at angular frequency \(\omega\)

Lagrangian Mechanics

Reformulates Newton’s laws using energy instead of force.

Define the Lagrangian \(L = T - V\) (kinetic \(-\) potential energy).

The Euler–Lagrange equation: \[\frac{d}{dt}\!\left(\frac{\partial L}{\partial \dot{q}}\right) - \frac{\partial L}{\partial q} = 0\]

For the SHO: \(T = \tfrac{1}{2}m\dot{x}^2\), \(V = \tfrac{1}{2}kx^2\), so \(L = \tfrac{1}{2}m\dot{x}^2 - \tfrac{1}{2}kx^2\).

\[\frac{\partial L}{\partial \dot{x}} = m\dot{x}, \quad \frac{\partial L}{\partial x} = -kx\]

\[\Rightarrow \quad \frac{d}{dt}(m\dot{x}) + kx = 0 \quad \Longrightarrow \quad m\ddot{x} + kx = 0 \checkmark\]

Hamiltonian Mechanics

Describes mechanics via total energy \(H = T + V\) in terms of position \(q\) and momentum \(p = m\dot{q}\).

Hamilton’s equations — a system of two 1st-order ODEs: \[\dot{q} = \frac{\partial H}{\partial p}, \qquad \dot{p} = -\frac{\partial H}{\partial q}\]

For the SHO: \(H = \dfrac{p^2}{2m} + \dfrac{1}{2}kq^2\)

\[\dot{q} = \frac{p}{m}, \qquad \dot{p} = -kq \quad \Rightarrow \quad \ddot{q} + \omega^2 q = 0 \checkmark\]

Tip

Any \(n\)th-order ODE can be converted to a system of \(n\) first-order ODEs. This is standard practice in both theory and numerical computation.

Three Frameworks — One ODE

Framework	Starting Point	Result
Newton	\(F = ma\)	\(m\ddot{x} + kx = 0\)
Lagrange	Euler–Lagrange eq.	\(m\ddot{x} + kx = 0\)
Hamilton	Hamilton’s eqs.	\(m\ddot{x} + kx = 0\)

The ODE is the fundamental mathematical object encoding the physics. Different frameworks are different roads to the same destination.

SHO: Solution and Phase Portrait

Figure 3: Left: position \(x(t)\). Right: phase portrait — each closed ellipse is a periodic solution.

Tip

The phase portrait encodes the full family of solutions. Closed curves = periodic motion.

Electrical Circuits

The RLC Circuit

Element	Voltage–Current Relation
Resistor \(R\)	\(V_R = RI\)
Capacitor \(C\)	\(I = C\,\dot{V}_C\)
Inductor \(L\)	\(V_L = L\,\dot{I}\)

Kirchhoff’s voltage law: \(V_L + V_R + V_C = E(t)\)

With \(I = \dot{Q}\) and \(V_C = Q/C\):

\[\boxed{L\ddot{Q} + R\dot{Q} + \frac{Q}{C} = E(t)}\]

2nd-order, linear, constant coefficients — identical in form to the driven oscillator!

Mechanical–Electrical Analogy

Mechanical	Electrical
Mass \(m\)	Inductance \(L\)
Damping \(b\)	Resistance \(R\)
Spring constant \(k\)	\(1/C\)
Force \(F(t)\)	EMF \(E(t)\)
Displacement \(x\)	Charge \(Q\)

RC circuit (no inductor, no drive): \(\quad R\dot{Q} + Q/C = 0 \quad\Rightarrow\quad \dot{Q} = -Q/(RC)\)

Population Dynamics

Exponential and Logistic Growth

Malthusian / Exponential growth (Malthus, 1798): \[\frac{dP}{dt} = rP, \quad P(0)=P_0 \quad\Rightarrow\quad P(t) = P_0 e^{rt}\]

Predicts unlimited growth (\(r>0\)) or extinction (\(r<0\)). Unrealistic long-term.

Logistic growth (Verhulst, 1838): \[\frac{dP}{dt} = rP\!\left(1-\frac{P}{K}\right), \quad P(0)=P_0\]

Exact solution: \(\displaystyle P(t) = \frac{K}{1+\left(\dfrac{K}{P_0}-1\right)e^{-rt}}\)

As \(t\to\infty\): \(P(t)\to K\) — the stable equilibrium (carrying capacity).

Logistic Growth: Solution Curves

Figure 5: Logistic growth (\(r=1\), \(K=100\)). All trajectories converge to \(K\) regardless of \(P_0\).

Tip

Solutions above \(K\) decrease; solutions below \(K\) increase. \(P=K\) is a stable equilibrium; \(P=0\) is unstable.

Predator–Prey: Lotka–Volterra

When two species interact we need a system of ODEs.

The Lotka–Volterra equations: \[\frac{dx}{dt} = \alpha x - \beta xy, \qquad \frac{dy}{dt} = \delta xy - \gamma y\]

\(x(t)\) = prey population; \(y(t)\) = predator population

Nonlinear (cross-product terms \(xy\))
No closed-form solution in general
Exhibits oscillatory behavior — we will return to this when we study systems of ODEs

Summary

Key Takeaways

A differential equation relates an unknown function to its derivatives. Solutions are functions, not numbers.
ODEs are classified by order, linearity, homogeneity, coefficient type, and autonomy.
An IVP pins down a unique solution by specifying initial data. The Picard–Lindelöf theorem guarantees existence and uniqueness under mild conditions.
Newton, Lagrange, and Hamilton all produce the same ODE for the simple harmonic oscillator — the ODE is the fundamental object.
ODEs appear everywhere: mechanics (\(F=ma\)), circuits (Kirchhoff), and biology (logistic growth, Lotka–Volterra).

Note

Next: Separable equations and exact solutions — Logan §1.2–1.3.

References

Logan, J David. 2015. A First Course in Differential Equations Third Edition.