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Differential Equations · Lessons

Differential Equations lessons

8 sections · 77 skills · work through the full outline skill by skill — read the long-form lesson or flip through the slide-style version.
Long-form lessonsSlide-style lessons

Differential Equations

Practice 77 skills across this outline.

Recommended next
Frobenius Method
Series and Numerical Methods · Power Series Solutions
Long-form lesson
Weighting distribution
22 categories
Series and Numerical Methods · Numerical Methods
10%
Problem-Solving & Modeling Tools · Method Selection and Workflow
8%
Foundations & Qualitative Analysis · Modeling and Solution Concepts
6%
Foundations & Qualitative Analysis · Slope Fields and Phase Analysis
6%
Laplace Transforms · Solving IVPs with Laplace
5%
Second-Order Linear ODEs · Mechanical Vibrations
5%

Skills

The comprehensive taxonomy of differential equations — drill into a card to work its skills.

Series and Numerical Methods · Power Series Solutions

3 core skills3% total weight
0/3 completed
3 skills to mastery
1
Frobenius MethodRecommended
Long-form
2
Series Solutions: Ordinary PointsAvailable
Long-form
3
Power Series MethodAvailable
Long-form

Series and Numerical Methods · Numerical Methods

4 core skills10% total weight
0/4 completed
4 skills to mastery
1
Euler's MethodAvailable
Long-form
2
Runge-Kutta MethodAvailable
Long-form
3
Technology for SolutionsAvailable
Long-form
4
Improved Euler/Heun's MethodAvailable
Long-form

First-Order Differential Equations · Separable Equations

4 core skills4% total weight
0/4 completed
4 skills to mastery
1
Solving Separable EquationsAvailable
Long-form
2
Identifying Separable DEsAvailable
Long-form
3
Initial Conditions: Separable EquationsAvailable
Long-form
4
Implicit vs. Explicit SolutionsAvailable
Long-form

First-Order Differential Equations · Linear First-Order Equations

4 core skills3% total weight
0/4 completed
4 skills to mastery
1
Linear ODEs with Initial ConditionsAvailable
Long-form
2
Integrating Factor MethodAvailable
Long-form
3
Identifying Linear First-Order ODEsAvailable
Long-form
4
Modeling with Linear ODEsAvailable
Long-form

First-Order Differential Equations · Substitution Methods

3 core skills3% total weight
0/3 completed
3 skills to mastery
1
Homogeneous First-Order EquationsAvailable
Long-form
2
Logistic Equation SolutionsAvailable
Long-form
3
Bernoulli EquationsAvailable
Long-form

First-Order Differential Equations · Exact Equations and Integrating Factors

3 core skills2% total weight
0/3 completed
3 skills to mastery
1
Integrating Factors: Non-ExactAvailable
Long-form
2
Solving Exact DEsAvailable
Long-form
3
Identifying Exact DEsAvailable
Long-form

Laplace Transforms · Piecewise and Impulse Inputs

3 core skills4% total weight
0/3 completed
3 skills to mastery
1
Dirac Delta FunctionsAvailable
Long-form
2
Heaviside Step FunctionsAvailable
Long-form
3
ODEs with Discontinuous InputsAvailable
Long-form

Laplace Transforms · Laplace Transform Basics

3 core skills4% total weight
0/3 completed
3 skills to mastery
1
Definition and Basic Transform RulesAvailable
Long-form
2
Linearity and Shifting TheoremsAvailable
Long-form
3
Transforms of Common FunctionsAvailable
Long-form

Laplace Transforms · Solving IVPs with Laplace

4 core skills5% total weight
0/4 completed
4 skills to mastery
1
Laplace of DerivativesAvailable
Long-form
2
Inverse Laplace via Partial FractionsAvailable
Long-form
3
Solving IVPs via LaplaceAvailable
Long-form
4
Convolution TheoremAvailable
Long-form

Second-Order Linear ODEs · Mechanical Vibrations

4 core skills5% total weight
0/4 completed
4 skills to mastery
1
Damping TypesAvailable
Long-form
2
Forced Oscillations & ResonanceAvailable
Long-form
3
Amplitude, Frequency & PhaseAvailable
Long-form
4
Mass-Spring SystemsAvailable
Long-form

Second-Order Linear ODEs · Nonhomogeneous Methods

3 core skills4% total weight
0/3 completed
3 skills to mastery
1
Undetermined CoefficientsAvailable
Long-form
2
Variation of ParametersAvailable
Long-form
3
Superposition & Particular SolutionsAvailable
Long-form

Second-Order Linear ODEs · Homogeneous Linear ODEs (Constant Coefficients)

4 core skills4% total weight
0/4 completed
4 skills to mastery
1
Repeated RootsAvailable
Long-form
2
Characteristic EquationAvailable
Long-form
3
Complex Roots & OscillationsAvailable
Long-form
4
Distinct Real RootsAvailable
Long-form

Systems of Differential Equations · Phase Plane and Qualitative Behavior

3 core skills5% total weight
0/3 completed
3 skills to mastery
1
Classifying Critical PointsAvailable
Long-form
2
Phase Portraits & TrajectoriesAvailable
Long-form
3
Stability in SystemsAvailable
Long-form

Systems of Differential Equations · Applications of Systems

3 core skills4% total weight
0/3 completed
3 skills to mastery
1
Coupled Mass-Spring SystemsAvailable
Long-form
2
Compartment ModelsAvailable
Long-form
3
Predator-Prey ModelsAvailable
Long-form

Systems of Differential Equations · Linear Systems and Matrix Methods

4 core skills4% total weight
0/4 completed
4 skills to mastery
1
Systems: Real Distinct EigenvaluesAvailable
Long-form
2
Systems in Matrix FormAvailable
Long-form
3
Systems: Complex EigenvaluesAvailable
Long-form
4
Eigenvalues & Eigenvectors: 2x2 SystemsAvailable
Long-form

Applications of First-Order ODEs · Physical and Thermal Models

3 core skills4% total weight
0/3 completed
3 skills to mastery
1
Falling Objects with DragAvailable
Long-form
2
Newton's Law of Cooling/HeatingAvailable
Long-form
3
RC Circuit ModelsAvailable
Long-form

Applications of First-Order ODEs · Growth, Decay, and Logistic Models

4 core skills5% total weight
0/4 completed
4 skills to mastery
1
Exponential Growth & DecayAvailable
Long-form
2
Growth Model ParametersAvailable
Long-form
3
Logistic Growth & Carrying CapacityAvailable
Long-form
4
Doubling Time & Half-LifeAvailable
Long-form

Applications of First-Order ODEs · Mixing and Related Models

3 core skills3% total weight
0/3 completed
3 skills to mastery
1
Setting Up DEs from Word ProblemsAvailable
Long-form
2
Mixing ProblemsAvailable
Long-form
3
Salt & Concentration ModelsAvailable
Long-form

Foundations & Qualitative Analysis · Modeling and Solution Concepts

4 core skills6% total weight
0/4 completed
4 skills to mastery
1
General vs. Particular SolutionsAvailable
Long-form
2
Intro to Differential EquationsAvailable
Long-form
3
Equilibrium Solutions & StabilityAvailable
Long-form
4
IVPs & Existence/UniquenessAvailable
Long-form

Foundations & Qualitative Analysis · Slope Fields and Phase Analysis

4 core skills6% total weight
0/4 completed
4 skills to mastery
1
Slope FieldsAvailable
Long-form
2
Stability from Phase LinesAvailable
Long-form
3
Phase Line AnalysisAvailable
Long-form
4
Qualitative Behavior AnalysisAvailable
Long-form

Problem-Solving & Modeling Tools · Method Selection and Workflow

4 core skills8% total weight
0/4 completed
4 skills to mastery
1
Classifying ODEsAvailable
Long-form
2
Common Setup PitfallsAvailable
Long-form
3
Verifying SolutionsAvailable
Long-form
4
Interpreting Solutions in ContextAvailable
Long-form

Problem-Solving & Modeling Tools · Graphical and Interpretation Skills

3 core skills5% total weight
0/3 completed
3 skills to mastery
1
Analytic vs. Numeric SolutionsAvailable
Long-form
2
Parameter SensitivityAvailable
Long-form
3
Direction Fields & IsoclinesAvailable
Long-form