Working archive

1000 Ordinary Differential Equations

A long-term archive of 1,000 ordinary differential equations, organized by structure, analytical solution method, verification, graphical behavior, computational visualization, interactive ODE models, and application-oriented takeaways.

Problems are adapted from textbooks, course materials, and generative AI. All written solutions and explanations are done by me unless noted otherwise.

Status60/1000 Entries complete

FormatWork, graph, model lab

Project scope

This is a technical project combining differential equations, analytical solutions, mathematical writing, computational visualization, and web development. The goal is to build a searchable, application-oriented reference system with selected interactive ODE models, not just a list of solved problems.

Completed work

Entries

Showing 1-60 of 60 entries

#001

y' = 2x

#002

y' = 3x^2

#003Model Lab

y' = 5x^4 - 2x + 7

#004

y' = x^{-2}

#005

y' = 4\sqrt{x}

#006

y' = \frac{1}{\sqrt{x}}

#007

y' = 6x^2 - 4x^{-1}

#008

y' = e^x

#009

y' = 2e^{2x}

#010

y' = e^{-3x}

#011

y' = \sin x

#012

y' = 4\cos x

#013

y' = 3\sec^2 x

#014

y' = 2\csc x\cot x

#015

y' = x^3 + e^x

#016

y' = 2x - \sin x

#017

y' = x^2 + 3\cos x - 4e^x

#018

y' = \frac{1}{x}

#019

y' = \frac{3}{2x + 1}

#020

y' = \frac{x}{x^2 + 1}

#021

y' = \frac{2x}{x^2 + 5}

#022

y' = \frac{1}{1 + x^2}

#023

y' = \frac{1}{\sqrt{1 - x^2}}

#024

y' = x^2e^{x^3}

#025

y' = 2x\cos(x^2)

Notes

Direct integration equations notes

#026

y' = 2x,\quad y(0)=3

#027

y' = 3x^2,\quad y(1)=5

#028Model Lab

y' = 5x^4 - 2x + 7,\quad y(0)=-1

#029

y' = x^{-2},\quad y(1)=0

#030

y' = 4\sqrt{x},\quad y(0)=2

#031

y' = \frac{1}{\sqrt{x}},\quad y(1)=4

#032

y' = 6x^2 - 4x^{-1},\quad y(1)=0

#033

y' = e^x,\quad y(0)=1

#034

y' = 2e^{2x},\quad y(0)=3

#035

y' = e^{-3x},\quad y(0)=0

#036

y' = \sin x,\quad y(0)=2

#037

y' = 4\cos x,\quad y(0)=-1

#038

y' = 3\sec^2 x,\quad y(0)=0

#039

y' = 2\csc x\cot x,\quad y(\pi/2)=1

#040

y' = 4x^3 - 6x + 2,\quad y(1)=0

#041

y' = 3\sqrt{x} + 2e^x,\quad y(0)=5

#042

y' = \frac{2x}{x^2 + 4},\quad y(0)=1

#043

y' = \frac{1}{x + 2},\quad y(0)=0

#044

y' = 5\cos(5x),\quad y(0)=2

#045

y' = 2x\sin(x^2),\quad y(0)=3

#046

y' = x^2e^{x^3},\quad y(0)=0

#047

y' = \frac{1}{1 + 4x^2},\quad y(0)=-1

#048

y' = \frac{3}{\sqrt{1 - 9x^2}},\quad y(0)=4

#049

y' = e^x + \cos x + 2x,\quad y(0)=0

#050

y' = \frac{x}{\sqrt{x^2 + 1}},\quad y(0)=2

Notes

Direct integration initial value problems notes

#051

y' = y

#052

y' = -2y

#053Model Lab

y' = 3y

#054

y' = y + 4

#055

y' = 5 - y

#056

y' = y^2

#057

y' = -y^2

#058

y' = y(1-y)

#059

y' = 2y(3-y)

#060

y' = y(y-4)

Taxonomy

Roadmap

I. Direct Integration and Basic First-Order ODEs

001-025: Direct integration equations, $y' = f(x)$
026-050: Direct integration initial value problems
051-080: Autonomous equations, $y' = f(y)$
081-100: Equilibrium solutions and phase-line analysis

II. Separable and Substitution-Based First-Order ODEs

101-150: Separable equations
151-180: Separable initial value problems
181-210: Homogeneous first-order equations, $y' = F(y/x)$
211-235: Bernoulli equations
236-260: Riccati equations with known particular solutions
261-280: Miscellaneous substitution methods

III. Linear and Exact First-Order ODEs

281-330: First-order linear equations
331-360: First-order linear initial value problems
361-410: Exact equations
411-435: Integrating factors for non-exact equations
436-460: Clairaut, Lagrange, and special first-order equations

IV. First-Order Modeling

461-485: Exponential growth and decay
486-510: Logistic growth
511-535: Newton cooling/heating
536-565: Mixing problems
566-590: Falling objects and drag
591-610: RC and RL circuits

V. Second-Order Linear Homogeneous ODEs

611-650: Distinct real characteristic roots
651-680: Repeated characteristic roots
681-720: Complex characteristic roots
721-745: Second-order initial value problems
746-760: Boundary value problems

VI. Second-Order Linear Nonhomogeneous ODEs

761-800: Undetermined coefficients, polynomial/exponential forcing
801-835: Undetermined coefficients, sinusoidal forcing
836-865: Mixed forcing and resonance
866-900: Variation of parameters

VII. Cauchy-Euler, Reduction of Order, and Higher-Order ODEs

901-925: Cauchy-Euler equations
926-945: Reduction of order
946-970: Higher-order linear constant-coefficient ODEs

VIII. Systems, Laplace, Numerical, and Boss Fights

971-980: Linear systems of ODEs
981-988: Matrix exponentials and eigenvalue methods
989-993: Laplace transform methods
994-997: Numerical methods: Euler and Runge-Kutta
998-1000: Nonlinear/boss-fight equations

Capabilities

Skills Demonstrated

Skill	Evidence in archive
Mathematical modeling	Classification by ODE type, solution method, and model behavior
Technical writing	Takeaways and explanations for each entry
Computational thinking	Graphs, derivative checks, computational visualization
Python / MATLAB	Code-backed plots and planned interactive ODE models
Web development	Interactive searchable archive
Long-term execution	1,000-entry roadmap with completed-entry tracking

Computational showcase

ODE Model Lab

3 model-lab entries planned

Selected entries are not only solved and archived — they are turned into interactive models with plots, sliders, and system-level explanations.

#003

Polynomial slope curve family

Planned model

Equation

$y' = 5x^4 - 2x + 7$

Model statusPlanned interactive model

Model categoryDirect integration

A visual model for connecting the polynomial slope field y' = 5x^4 - 2x + 7 to its shifted family of solution curves.

Simulation placeholderComing soon

interactive model surface

Interactive simulationPlotsSlidersWhy this mattersModel limitations

#028

Initial-value polynomial trajectory

Planned model

Equation

$y' = 5x^4 - 2x + 7,\quad y(0)=-1$

Model statusPlanned interactive model

Model categoryInitial value problems

A planned model for showing how the initial condition y(0) = -1 selects one curve from the polynomial solution family.

Simulation placeholderComing soon

interactive model surface

Interactive simulationPlotsSlidersWhy this mattersModel limitations

#053

Exponential growth sensitivity

Planned model

Equation

$y' = 3y$

Model statusPlanned interactive model

Model categoryAutonomous systems

A simple autonomous model for comparing initial conditions, equilibrium behavior, and rapid exponential sensitivity over time.

Simulation placeholderComing soon

interactive model surface

Interactive simulationPlotsSlidersWhy this mattersModel limitations

“Once you learn the concept of a differential equation, you see differential equations all over, no matter what you do.”

Gian-Carlo Rota

1000 Ordinary Differential Equations

Project scope

Tags

Method tags

Concept tags

Lab tags

Entries

Notes

Notes

Roadmap

I. Direct Integration and Basic First-Order ODEs

II. Separable and Substitution-Based First-Order ODEs

III. Linear and Exact First-Order ODEs

IV. First-Order Modeling

V. Second-Order Linear Homogeneous ODEs

VI. Second-Order Linear Nonhomogeneous ODEs

VII. Cauchy-Euler, Reduction of Order, and Higher-Order ODEs

VIII. Systems, Laplace, Numerical, and Boss Fights

Skills Demonstrated

ODE Model Lab

Polynomial slope curve family

Initial-value polynomial trajectory

Exponential growth sensitivity