Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate ↓


Differential Equations


A differential equation is a mathematical equation that involves functions and their derivatives. These equations are important in expressing various natural phenomena in science and engineering, such as physics, chemistry, biology, and economics.

What are differential equations?

Differential equations consist of an unknown function and its derivative. For example, the function may describe some physical phenomenon such as motion, heat, or bacterial growth, and its derivative may describe the rate of change of these phenomena.

Basic example

Consider the simple differential equation:

dy/dx = 3x

In this equation, dy/dx represents the derivative of y with respect to x, and it is equal to 3x. The objective is to find a function y that satisfies this equation.

Types of differential equations

Differential equations can be classified into several types:

  • Ordinary Differential Equations (ODE): These are equations with one independent variable. For example: 
    dy/dx + y = x
  • Partial Differential Equations (PDE): These have more than one independent variable. For example: 
    ∂u/∂t = ∂²u/∂x²

Orders and degrees

The order of a differential equation is the highest order of the derivatives present in the equation. The degree is the power of the highest derivative, provided the equation is a polynomial in its derivatives.

For example, in the differential equation:

d²y/dx² + 3(dy/dx)² = y

The order is 2 (since the highest derivative is d²y/dx²), and the degree is 1 (treating it as a polynomial in derivatives).

Solutions of differential equations

The solution of a differential equation is a function that satisfies the equation when substituted in it. Let us explore various methods to solve various differential equations.

Separable equation

The separating equation can be expressed as the product of two functions: one that involves only x and one that involves only y.

dy/dx = g(x)h(y)

To solve, isolate the variables:

dy/h(y) = g(x) dx

Then integrate both sides.

Example

Consider:

dy/dx = xy

Rearrange to isolate the variables:

dy/y = x dx

Integrate both sides:

∫(1/y) dy = ∫x dx

The solution is this:

ln|y| = (1/2)x² + C

where C is the constant of integration.

First-order linear differential equations

The general form of a first-order linear differential equation is:

dy/dx + P(x)y = Q(x)

To solve, find the integrating factor μ(x) which is defined as:

μ(x) = e^(∫P(x) dx)

Multiply the entire equation by μ(x) to make the left side the derivative of the product μ(x)y.

Example

Consider the equation:

dy/dx + 2y = x

Here, P(x) = 2 is the integrating factor:

μ(x) = e^(∫2 dx) = e^(2x)

Multiply by e^(2x):

e^(2x) dy/dx + 2e^(2x) y = xe^(2x)

Note that the left side is the derivative of e^(2x)y. Solve by integrating both sides:

d/dx (e^(2x)y) = xe^(2x)

Application of differential equations

Differential equations are used to model real-world problems.

Population growth

A simple model of population growth, in which the rate of change in the population is proportional to the population itself, is given as follows:

dp/dt = kp

Where p is the population, and k is the constant of proportionality. This can be solved by separation of variables such as:

dp/p = k dt

The solution is this:

p(t) = p₀e^(kt)

where p₀ is the initial population at t = 0.

Newton's law of cooling

This law states that the rate of change of temperature of an object is proportional to the difference between its temperature and the temperature of the surroundings. It is described by the difference equation:

dT/dt = -k(T - Tₐ)

where T is the temperature of the object, Tₐ is the temperature of the surroundings, and k is a positive constant. The solution is:

T(t) = Tₐ + (T₀ - Tₐ)e^(-kt)

Where T₀ is the initial temperature of the object.

Conclusion

Differential equations are a vast and important area of mathematics with a lot of applications. They help us model, solve, and understand various phenomena in our daily lives and scientific explorations. This introduction covers some of the basic concepts, types, solutions, and examples of differential equations widely used in undergraduate studies.


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