Solve differential equations and applications
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Differential Equations is available on both platforms
iPhone/iPad: download the single MasterNow app from the App Store — Differential Equations is included with all subjects. Android: download the dedicated Differential Equations subject app from Google Play.
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Sub Topics
192
MCQs
82
MCOs
141
True/False
80
Fill Blanks
35
Rearrange
81
Matching
41
Comprehensions
74
Flashcard Decks
Curriculum
What You'll Learn
01 Introduction to Differential Equations 3 topics
1 What are Differential Equations?
- Definition and Classification
- Order and Degree
- Linear vs. Nonlinear Equations
2 Modeling with Differential Equations
- Physical Systems
- Population Growth Models
- Electrical Circuits
- Mechanical Systems
3 Direction Fields and Solution Curves
- Constructing Direction Fields
- Interpreting Solution Behavior
- Equilibrium Solutions
02 First-Order Differential Equations 6 topics
1 Separable Equations
- Method of Separation of Variables
- Initial Value Problems
- Applications
2 Linear First-Order Equations
- Integrating Factor Method
- Homogeneous vs. Nonhomogeneous Equations
- Applications
3 Exact Equations and Integrating Factors
- Testing for Exactness
- Finding Potential Functions
- Creating Integrating Factors
4 Special Types of First-Order Equations
- Homogeneous Equations
- Bernoulli Equations
- Ricatti Equations
- Clairaut Equations
5 Substitution Methods
- Change of Variables
- Special Substitutions
6 Existence and Uniqueness Theorems
- Picard's Theorem
- Lipschitz Conditions
- Extended Existence Results
03 Applications of First-Order Differential Equations 5 topics
1 Growth and Decay Models
- Exponential Growth
- Logistic Growth
- Competition Models
2 Newton's Law of Cooling
- Applications
3 Mixture Problems
- Well-Mixed Assumption
- Series of Tanks
4 Trajectory Problems
- Orthogonal Trajectories
- Brachistochrone Problem
5 Financial Models
- Compound Interest
- Amortization
04 Higher-Order Linear Differential Equations 5 topics
1 Basic Theory
- Linear Operators
- Homogeneous vs. Nonhomogeneous Equations
- Linear Independence and Wronskian
2 Homogeneous Linear Equations with Constant Coefficients
- Characteristic Equation Method
- Real and Distinct Roots
- Repeated Roots
- Complex Roots
3 Reduction of Order
- When One Solution is Known
- Applications
4 Nonhomogeneous Linear Equations
- Method of Undetermined Coefficients
- Variation of Parameters
- Superposition Principle
5 Cauchy-Euler Equations
- Substitution Methods
- Solutions Using Complex Arithmetic
05 Applications of Higher-Order Differential Equations 4 topics
1 Spring-Mass Systems
- Free Undamped Motion
- Free Damped Motion
- Forced Motion
- Resonance
2 Electric Circuits
- RLC Circuits
- Transient Response
- Steady-State Response
3 Mechanical Vibrations
- Torsional Vibrations
- Coupled Oscillators
4 Beam Deflection Problems
- Euler-Bernoulli Beam Theory
- Boundary Value Problems
06 Systems of First-Order Linear Equations 5 topics
1 Introduction to Systems
- Conversion from Higher-Order to Systems
- Matrix Notation
2 Homogeneous Linear Systems with Constant Coefficients
- Eigenvalue Method
- Real Distinct Eigenvalues
- Complex Eigenvalues
- Repeated Eigenvalues
3 Nonhomogeneous Linear Systems
- Method of Undetermined Coefficients
- Variation of Parameters
4 Matrix Exponential
- Definition and Properties
- Computing the Matrix Exponential
- Solving Systems Using Matrix Exponential
5 Stability Analysis
- Equilibrium Points
- Stability Criteria
- Phase Portraits
07 Nonlinear Differential Equations and Stability 5 topics
1 Autonomous Systems
- Phase Plane Analysis
- Critical Points and Stability
2 Linearization
- Jacobian Matrix
- Linear Approximation Near Equilibrium
3 Classification of Critical Points
- Nodes, Saddles, and Spirals
- Centers and Limit Cycles
4 Lyapunov Stability
- Lyapunov Functions
- Asymptotic Stability
5 Bifurcation Theory
- Saddle-Node Bifurcations
- Hopf Bifurcations
- Pitchfork Bifurcations
08 Power Series Solutions 4 topics
1 Series Solutions Near Ordinary Points
- Power Series Method
- Radius of Convergence
- Recurrence Relations
2 Regular Singular Points
- Frobenius Method
- Indicial Equation
- Equal Roots of the Indicial Equation
- Differing by an Integer
3 Bessel's Equation
- Bessel Functions of the First Kind
- Bessel Functions of the Second Kind
- Modified Bessel Functions
- Applications
4 Legendre's Equation
- Legendre Polynomials
- Rodrigues' Formula
- Orthogonality Properties
- Applications in Physics
09 Laplace Transform Methods 6 topics
1 Definition and Properties
- Basic Transforms
- Operational Properties
- Transform of Derivatives
2 Inverse Laplace Transform
- Partial Fraction Decomposition
- Convolution Theorem
3 Solving Initial Value Problems
- Step-by-Step Procedure
4 Unit Step and Dirac Delta Functions
- Heaviside Function
- Shifted Functions
- Impulse Functions
5 Systems of Differential Equations
- Transfer Functions
- Block Diagrams
6 Convolution
- Duhamel's Formula
- Applications
10 Fourier Series and Partial Differential Equations 6 topics
1 Fourier Series
- Fourier Coefficients
- Even and Odd Functions
- Convergence Theorems
2 Sturm-Liouville Problems
- Eigenvalue Problems
- Orthogonal Eigenfunctions
- Expansion in Eigenfunctions
3 Introduction to Partial Differential Equations
- Classification of PDEs
- Initial and Boundary Conditions
4 Heat Equation
- Derivation
- Separation of Variables
- Initial-Boundary Value Problems
5 Wave Equation
- Derivation
- D'Alembert's Solution
- Vibrating String Problems
6 Laplace's Equation
- Harmonic Functions
- Boundary Value Problems
- Applications
11 Numerical Methods for Differential Equations 5 topics
1 Euler's Method
- Forward Euler
- Backward Euler
- Error Analysis
2 Runge-Kutta Methods
- Second-Order Methods
- Fourth-Order Methods
- Adaptive Step Size
3 Multistep Methods
- Adams-Bashforth Methods
- Adams-Moulton Methods
- Predictor-Corrector Methods
4 Boundary Value Problems
- Shooting Methods
- Finite Difference Methods
5 Stability and Stiffness
- Stability Analysis
- Stiff Differential Equations
- Implicit Methods
12 Special Topics in Differential Equations 5 topics
1 Green's Functions
- Construction of Green's Functions
- Applications to Boundary Value Problems
2 Dynamical Systems Theory
- Poincaré Maps
- Chaos and Strange Attractors
- Fractals
3 Delay Differential Equations
- Basic Theory
- Stability Analysis
- Applications
4 Stochastic Differential Equations
- Introduction to Stochastic Processes
- Itô's Formula
- Applications in Finance
5 Control Theory
- Controllability and Observability
- Feedback Control
- Optimal Control Problems
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