bythreedu · Math Tracks · Curriculum v1.0 · 2026

Modeling change,
solving systems, and the
mathematics of
dynamics.

A complete ordinary differential equations course from first-order ODEs through systems and phase plane analysis. Built for students in math, physics, and engineering who need to model dynamic systems and understand their behavior. Every method is derived, every solution is interpreted.

Duration~18weeks
Total Lessons17at your pace
Units6progressive phases
Assignments6+problem sets
DestinationApplied Math Ready
00Approach
Course Philosophy

Modeling Is the Goal

Differential equations are the language of science and engineering. This course starts with the physical or geometric situation and derives the equation from it — not the other way around. Understanding where an ODE comes from makes solving it meaningful.

Qualitative Before Quantitative

Not every differential equation has a closed-form solution. This course teaches qualitative analysis — phase lines, phase planes, stability — so students can understand the behavior of a system without an explicit formula. That's the real skill.

01Curriculum
Learning Path
01
Unit 1 · Weeks 1–2
Introduction and First-Order ODEs
3 lessons · Classification · Separable equations · Linear equations
order and linearityseparable equationsintegrating factor
+
L1
What Is a Differential Equation?
Classification by order, linearity, and number of variables. Solutions as functions, not numbers. The general and particular solution. Initial value problems.
L2
Separable Equations
Separating variables and integrating both sides. Implicit solutions. Finding the particular solution from an initial condition. Modeling with separable equations.
L3
First-Order Linear Equations
Standard form. The integrating factor method derived from the product rule. Why the method works. Applications to mixing and cooling problems.
Assignment 1
Solve 10 first-order ODEs — 5 separable, 5 linear. Find the particular solution for each from a given initial condition. Model one mixing problem from start to finish.
You solve any first-order ODE and interpret the solution in context.
02
Unit 2 · Weeks 3–4
Autonomous Equations and Qualitative Analysis
2 lessons · Phase portraits · Equilibria · Stability
equilibrium solutionsstabilityphase line
+
L4
Autonomous Equations and Phase Lines
What makes an equation autonomous. Finding equilibrium solutions. The phase line as a tool for understanding long-term behavior without solving.
L5
Stability of Equilibria
Stable, unstable, and semi-stable equilibria from the phase line. Linearization near an equilibrium. The logistic equation as the central example.
Assignment 2
Draw the phase line for 5 autonomous equations. Classify each equilibrium. Describe the long-term behavior for initial conditions in each region.
You analyze the qualitative behavior of any autonomous ODE without solving it.
03
Unit 3 · Weeks 5–7
Second-Order Linear ODEs
3 lessons · Homogeneous equations · Characteristic equation · Particular solutions
characteristic equationundetermined coefficientsvariation of parameters
+
L6
Homogeneous Second-Order Linear ODEs
The characteristic equation. Three cases: two real roots, repeated root, complex conjugate roots. General solution as a linear combination.
L7
Undetermined Coefficients
Guessing the form of the particular solution based on the right-hand side. When the guess must be modified (resonance). The method of superposition.
L8
Variation of Parameters
A general method for particular solutions when undetermined coefficients fails. Derived from the Wronskian and the structure of the general solution.
Assignment 3
Solve 10 second-order linear ODEs — 5 homogeneous, 5 non-homogeneous (mix of methods). Show the complementary and particular solution for each non-homogeneous problem.
You solve any second-order linear ODE with constant coefficients.
04
Unit 4 · Weeks 8–9
Applications of Second-Order ODEs
2 lessons · Mechanical vibrations · Electrical circuits
underdampingresonanceRLC circuits
+
L9
Mechanical Vibrations
Free and forced oscillations. Underdamped, critically damped, overdamped. Resonance — what it means and why it matters. Mass-spring systems from Newton's second law.
L10
Electrical Circuits
The RLC circuit as the electrical analogue of the mechanical system. Kirchhoff's laws leading to a second-order ODE. Steady-state and transient responses.
Assignment 4
Model and solve 3 vibration problems and 2 circuit problems from physical descriptions. Classify each as underdamped, critically damped, or overdamped before solving.
You translate a physical system into a second-order ODE and interpret the solution.
05
Unit 5 · Weeks 10–12
Laplace Transforms
3 lessons · Transform definition · Inverse transforms · Solving IVPs
Laplace transformpartial fractionsstep functions
+
L11
The Laplace Transform
Definition as an integral transform. Transforms of standard functions. Linearity and the first shifting theorem.
L12
Inverse Laplace Transforms
Using partial fractions to invert. The second shifting theorem and Heaviside step functions. Dealing with discontinuous forcing functions.
L13
Solving IVPs with Laplace Transforms
The transform method as an algebraic alternative to the variation of parameters method. Convolution and the convolution theorem.
Assignment 5
Solve 8 initial value problems using Laplace transforms — including 3 with discontinuous forcing functions. Compare the Laplace method to undetermined coefficients on one problem.
You solve IVPs using Laplace transforms and understand when the method is advantageous.
06
Unit 6 · Weeks 13–18
Systems of ODEs and Capstone
4 lessons + capstone · Linear systems · Phase plane · Nonlinear systems
eigenvalue methodphase planelinearization near equilibria
+
L14
Systems of Linear ODEs
Matrix form. The eigenvalue method for 2x2 systems. Real distinct, repeated, and complex eigenvalue cases.
L15
Phase Plane Analysis
Trajectories as solution curves in the phase plane. Classifying equilibria of linear systems: node, saddle, spiral, center.
L16
Nonlinear Systems
Linearization near equilibria. When the linearization predicts the behavior of the nonlinear system and when it doesn't. Predator-prey as the central example.
L17 (cap)
Capstone: Exam Simulation and Modeling Project
A full-length differential equations exam simulation, plus a short modeling project of the student's choosing. Every missed exam problem corrected with reasoning.
Assignment 6
A full-length exam simulation across all 6 units, plus a modeling project connecting differential equations to a real phenomenon of your choice.
You complete differential equations with the ability to model, solve, and interpret any standard ODE system.
02Reference
Core Concepts
y'=f(x,y)
Classification
ODEs are classified by order (highest derivative), linearity (linear vs. nonlinear), and whether they are autonomous (right side depends only on y, not x). The classification determines which methods apply.
Order · Linearity · Autonomous
μ(x)
Integrating Factor
A first-order linear ODE in standard form y' + P(x)y = Q(x) is solved by multiplying by μ(x) = e^(∫P dx). This turns the left side into the derivative of a product.
Standard form · General solution · IVP
r²+br+c=0
Characteristic Equation
For a second-order linear ODE with constant coefficients, the characteristic equation determines the form of the general solution: two real roots, repeated root, or complex conjugate roots.
Undetermined coefficients · Variation of parameters
λ
Eigenvalues in Systems
For a linear system x' = Ax, the eigenvalues of A determine the behavior of solutions: real distinct → node or saddle, complex → spiral or center, repeated → degenerate node.
Phase plane · Stability · Linearization
L{f}
Laplace Transform
The Laplace transform converts a differential equation into an algebraic equation. It is especially useful for discontinuous forcing functions. L{f'} = sL{f} - f(0) connects the transform to initial conditions.
Step functions · Convolution · Inverse transform
dy/dt = ky
Exponential Models
The simplest ODE y' = ky has solution y = Ce^(kt). This models population growth, radioactive decay, and Newton's law of cooling. Most first-order linear models reduce to a variation of this.
Logistic growth · Mixing problems · Cooling
03Capstone
Exam Simulation and Modeling Project

The capstone combines a full-length exam simulation with a short modeling project. The modeling project asks you to take a real phenomenon, formulate the differential equation, solve it, and interpret the result. The exam covers every analytic technique in the course.

○
Exam Simulation
Full-length, timed, all 6 units.
○
Modeling Project
A real phenomenon of your choosing, modeled and analyzed from start to finish.
○
Written Corrections
Every missed exam problem corrected with full reasoning.
○
Applied Math Readiness
A final session on what PDEs, numerical methods, or engineering applications will require.
04Investment
Pricing Options
per Hour
$85
1-hr lesson
Flexible · No commitment
Online only
  • Flexible scheduling
  • 1-on-1 focused attention
  • No long-term commitment

Best for trying the curriculum before committing.

Book a session
Popular
per Month
$650
4 lessons · 1.5 hours/lesson
1 lesson/week · book more to move faster
Online + In-Person available
  • Session notes after every lesson
  • Async DM support between sessions
  • Priority scheduling

Booking 2x/week draws down lessons faster — no extra charge.

Get started
Semester
$3,775
26 lessons · 1.5 hours/lesson
~6 months · save vs. monthly
Online + In-Person available
  • Full course arc
  • Session notes after every lesson
  • Capstone review + corrections
  • Next track readiness session

The full arc in one commitment.

Enroll now
In-person sessions available on Monthly and Semester plans at $800/mo and $4,625/semester. NYC only. Per-hour sessions are online only.
05Contact
Let's Talk

Starting differential equations or need a structured pass through a difficult course? One session is enough to establish where you are and map the path forward.

What do I need coming in? +
Solid Calc II (integration techniques) and comfort with matrices from linear algebra or pre-req calculus. Calc III helps but is not always required depending on your program.
Is this for math majors or engineering students? +
Both. The course covers the core ODE content that appears in both math and engineering differential equations courses. The emphasis adjusts to who is in the room.
Are lessons online or in-person? +
Per-hour sessions are online only. Monthly and Semester plans include the option for in-person sessions in NYC.
What if I need to reschedule? +
Give 24 hours notice and we'll find another time. Monthly and Semester students get priority rebooking.