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Differential Equations

y'

Differential Equations

First-order, higher-order, systems, Laplace transforms, and PDEs

176Modules

10Units

Recommended preparation

Calculus 2 is the key preparation — you will integrate often and use techniques like partial fractions throughout.

Population DynamicsINTRO

The Language of ChangeLESSON

Verifying a Solution to dy/dx = 2xEXAMPLE

Classification ChallengePRACTICE

Differential Equations in NatureREAL-WORLD

Understanding Direction FieldsLESSON

Sketching Solutions ThroughEXAMPLE

Matching Fields to EquationsPRACTICE

Slope Field ExplorerACTIVITY

Direction Field LaboratoryACTIVITY

Initial Conditions and UniquenessLESSON

Introduction SummarySUMMARY

Separating Variables VisuallyINTRO

The Separation MethodLESSON

Solving dy/dx = y²xEXAMPLE

Separation WorkshopACTIVITY

Separable or Not?PRACTICE

The Integrating Factor MagicINTRO

Linear First-Order TheoryLESSON

Solving y' + 2y = e^xEXAMPLE

Integrating Factor CalculatorACTIVITY

Linear Equation MasteryPRACTICE

Conservative Vector FieldsINTRO

Exact Equation TheoryLESSON

Solving (2xy + 3)dx + (x² - 1)dy = 0EXAMPLE

Exactness DetectorACTIVITY

Testing and Solving Exact EquationsPRACTICE

Growth and Decay ProblemsREAL-WORLD

Mixing ProblemsREAL-WORLD

First-Order Methods SummarySUMMARY

First-Order ReferenceFORMULA SHEET

Growth Rate ExplorerINTRO

Exponential ModelsLESSON

Finding Doubling TimeEXAMPLE

Exponential Application ProblemsPRACTICE

Compound Interest CalculatorACTIVITY

Carbon Dating Ancient ArtifactsREAL-WORLD

Temperature DynamicsINTRO

Cooling and HeatingLESSON

Coffee Cooling ProblemEXAMPLE

Temperature ProblemsPRACTICE

Thermal Equilibrium LabACTIVITY

Murder Mystery TimelineREAL-WORLD

Beyond ExponentialINTRO

The Logistic EquationLESSON

Solving a Logistic IVPEXAMPLE

Population ModelingPRACTICE

Harvesting ModelsREAL-WORLD

Population SimulatorACTIVITY

Applications SummarySUMMARY

Applications AssessmentASSESSMENT

From First to Second OrderINTRO

The Characteristic EquationLESSON

Solving y'' - 5y' + 6y = 0EXAMPLE

Characteristic Root VisualizerACTIVITY

Finding General SolutionsPRACTICE

Guessing Particular SolutionsINTRO

Undetermined Coefficients TheoryLESSON

Solving y'' + 4y = 3sinEXAMPLE

Undetermined Coefficients PracticePRACTICE

Particular Solution FinderACTIVITY

When Guessing FailsINTRO

Variation of Parameters MethodLESSON

Solving y'' + y = secEXAMPLE

General Forcing FunctionsPRACTICE

Wronskian CalculatorACTIVITY

Spring-Mass SystemsREAL-WORLD

RLC Circuit AnalysisREAL-WORLD

Higher-Order SummarySUMMARY

Higher-Order ReferenceFORMULA SHEET

Phase Space ExplorerINTRO

Coupled TanksINTRO

From Scalar to VectorLESSON

Converting y'' + 3y' + 2y = 0EXAMPLE

System FormulationPRACTICE

System ConverterACTIVITY

Eigenvector GeometryINTRO

Solving Linear SystemsLESSON

Solving x' = [1,2;3,2]xEXAMPLE

Eigenvalue Solution MethodPRACTICE

Eigenvalue Calculator for SystemsACTIVITY

Classification of Critical PointsLESSON

Analyzing a Saddle PointEXAMPLE

Phase Portrait ClassificationPRACTICE

Phase Portrait StudioACTIVITY

Predator-Prey DynamicsREAL-WORLD

Coupled Spring-MassREAL-WORLD

Nonlinear Stability AnalysisREAL-WORLD

Systems SummarySUMMARY

Systems AssessmentASSESSMENT

Transform Solution MethodINTRO

Transform VisualizationINTRO

Definition and Basic TransformsLESSON

Computing L{t²e⁻³ᵗ}EXAMPLE

Basic Transform ComputationPRACTICE

The Transform MethodLESSON

Solving y'' + 4y = e^(-t), y(0) = 1, y'(0) = 0EXAMPLE

Laplace Equation SolverACTIVITY

Transform Solution ProblemsPRACTICE

Transform Properties LabINTRO

Operational PropertiesLESSON

Using First Shifting TheoremEXAMPLE

Transform Properties PracticePRACTICE

Property Application WorkshopACTIVITY

Unit Step and Impulse FunctionsREAL-WORLD

Signal ProcessingREAL-WORLD

Transform CalculatorACTIVITY

Laplace Methods SummarySUMMARY

Laplace Transform TablesFORMULA SHEET

Series Solution BuilderINTRO

Taylor Series ConvergenceLESSON

Series Method at Ordinary PointsLESSON

Solving y'' - xy = 0EXAMPLE

Power Series PracticePRACTICE

Recurrence Relation SolverACTIVITY

Singular PointsINTRO

Regular Singular PointsLESSON

Frobenius Method CalculatorACTIVITY

Frobenius ApplicationsPRACTICE

Logarithmic CasesLESSON

Bessel Function GalleryINTRO

Bessel's EquationLESSON

Solving x²y'' + xy' + (x² - 1/4)y = 0EXAMPLE

Special Function VisualizerACTIVITY

Legendre PolynomialsLESSON

Vibrating DrumheadREAL-WORLD

Special Functions in PhysicsREAL-WORLD

Special Function PropertiesPRACTICE

Series Solutions SummarySUMMARY

Series Methods ReferenceFORMULA SHEET

Step by StepINTRO

Euler's Method TheoryLESSON

Improved Euler MethodLESSON

Implementing Euler for y' = x + yEXAMPLE

Euler Variants PracticePRACTICE

Euler Method LaboratoryACTIVITY

Fourth-Order MagicINTRO

Runge-Kutta TheoryLESSON

RK4 for Predator-Prey SystemEXAMPLE

RK4 Implementation WorkshopACTIVITY

Method ComparisonPRACTICE

Adaptive Step SizeREAL-WORLD

Shooting MethodINTRO

Numerical BVP MethodsLESSON

Solving y'' = -y with y(0) = 0, y(π) = 1EXAMPLE

BVP SolverACTIVITY

Heat Equation Steady StateREAL-WORLD

Numerical Methods SummarySUMMARY

Numerical Methods AssessmentASSESSMENT

From ODEs to PDEsINTRO

Classification of PDEsLESSON

Classify the PDEPRACTICE

PDE ExplorerACTIVITY

Product SolutionsINTRO

Separation MethodLESSON

Separation of Variables PracticePRACTICE

Separation WorkshopACTIVITY

Building WavesINTRO

Fourier TheoryLESSON

Finding Fourier Series of f(x) = x on [-π,π]EXAMPLE

Fourier Series ApplicationsPRACTICE

Fourier Coefficient CalculatorACTIVITY

Vibrating StringREAL-WORLD

Signal AnalysisREAL-WORLD

Heat Equation with Initial TemperatureEXAMPLE

Heat Equation on [0,L]EXAMPLE

Verifying SolutionsEXAMPLE

PDEs SummarySUMMARY

PDE ReferenceFORMULA SHEET

PDEs AssessmentASSESSMENT

Engineering Applications ShowcaseREAL-WORLD

Comprehensive Problem SetPRACTICE

Method Selection FlowchartACTIVITY

Course SummarySUMMARY

Course Formula CompilationFORMULA SHEET

Final Comprehensive AssessmentASSESSMENT