bythreedu · Math Tracks · Curriculum v1.0 · 2026

Models, solutions,
and the mathematics
of change.

A complete Differential Equations course from first-order ODEs through systems and phase plane analysis. Built for students in STEM who need to model, solve, and interpret dynamic systems — not just manipulate symbols.

Total Lessons17at your pace
Units6progressive phases
Assignments6+problem sets
DestinationAdvanced STEM
Ready
00Approach
Course Philosophy

Solutions Are Functions

The most important conceptual shift in differential equations is that solutions are functions, not numbers. Every method in this course is framed around what kind of function solves a given equation and why — not just how to apply the procedure.

Qualitative Before Quantitative

Phase lines and phase planes give you the behavior of a system before you solve anything. This course introduces qualitative analysis early and returns to it throughout — because in real applications, knowing the long-term behavior often matters more than the exact formula.

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
+
02
Unit 2 · Weeks 3–4
Autonomous Equations and Qualitative Analysis
2 lessons · Phase portraits · Equilibria · Stability
equilibrium solutionsstabilityphase line
+
03
Unit 3 · Weeks 5–7
Second-Order Linear ODEs
3 lessons · Homogeneous equations · Characteristic equation · Particular solutions
characteristic equationundetermined coefficientsvariation of parameters
+
04
Unit 4 · Weeks 8–9
Applications of Second-Order ODEs
2 lessons · Mechanical vibrations · Electrical circuits
underdampingresonanceRLC circuits
+
05
Unit 5 · Weeks 10–12
Laplace Transforms
3 lessons · Transform definition · Inverse transforms · Solving IVPs
Laplace transformpartial fractionsstep functions
+
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
+
02Capstone
Final Exam Simulation

The capstone is a full-length exam simulation across all 6 units, plus a short modeling project connecting differential equations to a real physical or biological system of your choosing.

○
Gap Review
Two weakest units revisited before the exam simulation.
○
Timed Final Exam
Full-length, independent, same conditions as a real exam.
○
Written Corrections
Every missed problem corrected with reasoning — not just the right answer.
○
Advanced STEM courses Readiness Review
A final session on what Advanced STEM courses will demand from day one.
03Investment
Pricing Options
Location
per Session
$125
1.5 hrs · online only
Session length
  • —Flexible scheduling
  • —1-on-1 focused attention
  • —No long-term commitment

Best for trying the curriculum before committing to a plan.

Popular
per Month
$500
4 sessions · 1.5 hrs/session
Frequency
Session length
  • —Session notes after every lesson
  • —Async DM support between sessions
  • —Priority scheduling

Adjust frequency and session length above — price updates automatically.

Per Semester
$2,000
16 sessions · 1.5 hrs/session
Term length
Frequency
Session length
  • —All 6 units — full course arc
  • —Session notes after every lesson
  • —Capstone exam simulation + written corrections
  • —Advanced STEM courses readiness review at completion

The full Differential Equations arc in one commitment.

📍
In-person available · NYC onlyMonthly and semester plans can be done in-person. Select “In-person · NYC” above to see adjusted pricing.
04Contact
Let's Talk

Taking diff eq in college and hitting a wall? One session is enough to find where the breakdown is and map a path forward.

What do I need to know coming in?+
Are lessons online or in-person?+
Can I switch plans later?+
What if I need to reschedule?+