Differential Equations

Differential equations describe how quantities change and are fundamental to modelling physical phenomena. They relate functions to their derivatives, appearing throughout physics, engineering, biology, economics, and beyond.

Ordinary Differential Equations (ODEs)

An ODE involves functions of a single independent variable and their derivatives.

Classification

• Order: The highest derivative present (first-order, second-order, etc.)
• Linear vs Nonlinear: Linear if the dependent variable and its derivatives appear only to the first power
• Homogeneous vs Nonhomogeneous: Homogeneous if equal to zero; nonhomogeneous if equal to some function
• Autonomous vs Non-autonomous: Autonomous if the independent variable doesn’t appear explicitly

First-Order ODEs

Separable Equations

When the equation can be written as:

$\frac{dy}{dx}=f(x)g(y)$

Solve by separating variables:

$\int \frac{dy}{g(y)}=\int f(x)dx$

Example: $\frac{dy}{dx}=xy$ separates to $\int \frac{dy}{y}=\int xdx$, giving $y=C{e}^{x^2/2}$

Linear First-Order

Standard form:

$\frac{dy}{dx}+P(x)y=Q(x)$

Integrating factor: $\mu (x)=e^{\int P(x)dx}$

Multiply through by $\mu (x)$ to get:

$\frac{d}{dx}[\mu (x)y]=\mu (x)Q(x)$

Then integrate both sides.

Exact Equations

An equation $M(x,y)dx+N(x,y)dy=0$ is exact if:

$\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$

The solution is found by integrating $M$ with respect to $x$ and $N$ with respect to $y$, combining to find a potential function $F(x,y)=C$.

Bernoulli Equations

$\frac{dy}{dx}+P(x)y=Q(x)y^n$

Substitute $v=y^{1-n}$ to transform into a linear equation.

Second-Order Linear ODEs

Homogeneous with Constant Coefficients

$a{y}^{\prime \prime }+b{y}^{\prime }+cy=0$

Characteristic equation: $a{r}^2+br+c=0$

Discriminant	Roots	General Solution
$b^2-4ac>0$	Distinct real $r_1,r_2$	$y=C_1{e}^{r_{1}x}+C_2{e}^{r_{2}x}$
$b^2-4ac=0$	Repeated $r$	$y=(C_1+C_{2}x)e^{rx}$
$b^2-4ac<0$	Complex $\alpha \pm \beta i$	$y=e^{\alpha x}(C_1\cos \beta x+C_2\sin \beta x)$

Nonhomogeneous Equations

$a{y}^{\prime \prime }+b{y}^{\prime }+cy=g(x)$

General solution: $y=y_h+y_p$ (homogeneous + particular)

Method of Undetermined Coefficients
Guess the form of $y_p$ based on $g(x)$:

• $g(x)=e^{kx}$ → try $y_p=A{e}^{kx}$
• $g(x)=\sin (kx)$ or $\cos (kx)$ → try $y_p=A\cos (kx)+B\sin (kx)$
• $g(x)=x^n$ → try polynomial of degree $n$

Variation of Parameters
For $y^{\prime \prime }+p(x)y^{\prime }+q(x)y=g(x)$ with known solutions $y_1,y_2$ to the homogeneous equation:

$y_p=-y_1\int \frac{y_{2}g}{W}dx+y_2\int \frac{y_{1}g}{W}dx$

where $W=y_1{y}_2^{\prime }-y_2{y}_1^{\prime }$ is the Wronskian.

Systems of ODEs

Matrix form: ${\mathbf{x}}^{\prime }=A\mathbf{x}$

Eigenvalue Method

1. Find eigenvalues $\lambda$ from $\det (A-\lambda I)=0$
2. Find corresponding eigenvectors $\mathbf{v}$
3. General solution: $\mathbf{x}=C_1{e}^{{\lambda }_{1}t}{\mathbf{v}}_1+C_2{e}^{{\lambda }_{2}t}{\mathbf{v}}_2+\cdots$

Phase Portraits

• Real eigenvalues: nodes (stable if negative, unstable if positive)
• Complex eigenvalues: spirals (stable if negative real part) or centres (pure imaginary)
• Saddle points: one positive, one negative eigenvalue

Series Solutions

Power Series Solutions

Assume $y={\sum }_{n=0}^{\infty }{a}_n{x}^n$ and substitute into the ODE to find recurrence relations for coefficients.

Frobenius Method

For equations with regular singular points, assume:

$y=x^r{\sum }_{n=0}^{\infty }{a}_n{x}^n$

Find the indicial equation to determine $r$.

Singular Points

• Ordinary point: Coefficients are analytic
• Regular singular point: $(x-x_0)P(x)$ and $(x-x_0{)}^{2}Q(x)$ are analytic
• Irregular singular point: Neither condition holds

Partial Differential Equations (PDEs)

A PDE involves functions of multiple independent variables and their partial derivatives.

Classification

By Order and Linearity

• Order: highest partial derivative
• Linear: dependent variable and derivatives appear linearly
• Quasilinear: linear in highest-order derivatives

Second-Order Linear PDEs
For $A{u}_{xx}+B{u}_{xy}+C{u}_{yy}+\cdots =0$:

Type	Discriminant	Prototype
Elliptic	$B^2-4AC<0$	Laplace
Parabolic	$B^2-4AC=0$	Heat
Hyperbolic	$B^2-4AC>0$	Wave

The Big Three

Heat Equation (Parabolic)

$\frac{\partial u}{\partial t}={\alpha }^2{\nabla }^{2}u$

Properties:

• Models heat diffusion, chemical diffusion, probability
• Smoothing effect: discontinuities instantly smooth out
• Maximum principle: extrema occur on boundary or initial data

1D Solution via Separation of Variables
Assume $u(x,t)=X(x)T(t)$:

$\frac{T^{\prime }}{T}={\alpha }^2\frac{X^{\prime \prime }}{X}=-\lambda$

Yields eigenvalue problems with solutions depending on boundary conditions.

Wave Equation (Hyperbolic)

$\frac{{\partial }^{2}u}{\partial t^2}=c^2{\nabla }^{2}u$

Properties:

• Models vibrating strings, membranes, sound, light
• Finite propagation speed $c$
• Preserves discontinuities

D’Alembert’s Solution (1D)
$u(x,t)=\frac{1}{2}[f(x+ct)+f(x-ct)]+\frac{1}{2c}{\int }_{x-ct}^{x+ct}g(s)ds$

where $f$ is the initial displacement and $g$ is the initial velocity.

Laplace’s Equation (Elliptic)

${\nabla }^{2}u=\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}+\cdots =0$

Properties:

• Steady-state heat distribution, electrostatics, fluid flow
• Solutions are harmonic functions
• Mean value property: value at a point equals average over any surrounding sphere
• Maximum principle: no interior extrema

Solution Methods

Separation of Variables

1. Assume product solution: $u(x,t)=X(x)T(t)$
2. Substitute into PDE
3. Separate into ODEs with separation constant
4. Solve each ODE with boundary conditions
5. Combine via superposition

Fourier Series

Any “reasonable” function on $[0,L]$ can be expanded:

$f(x)=\frac{a_0}{2}+{\sum }_{n=1}^{\infty }\left[a_n\cos \frac{n\pi x}{L}+b_n\sin \frac{n\pi x}{L}\right]$

Coefficients:
$a_n=\frac{2}{L}{\int }_0^{L}f(x)\cos \frac{n\pi x}{L}dx,b_n=\frac{2}{L}{\int }_0^{L}f(x)\sin \frac{n\pi x}{L}dx$

Fourier Transform

For problems on infinite domains:

$\hat{f}(\omega )={\int }_{-\infty }^{\infty }f(x)e^{-i\omega x}dx$

$f(x)=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }\hat{f}(\omega )e^{i\omega x}d\omega$

Derivatives transform to multiplication: $\widehat{f^{\prime }}=i\omega \hat{f}$

Boundary and Initial Conditions

Type	Condition	Example
Dirichlet	Value specified	$u(0,t)=0$
Neumann	Normal derivative specified	$\frac{\partial u}{\partial n}=0$
Robin	Linear combination	$u+\alpha \frac{\partial u}{\partial n}=0$
Periodic	$u(0)=u(L)$	Circular domain

Well-Posedness (Hadamard): A problem is well-posed if a solution exists, is unique, and depends continuously on the data.

Numerical Methods

For ODEs

Euler’s Method
$y_{n+1}=y_n+hf(t_n,y_n)$

Simple but first-order accurate. Error proportional to $h$.

Runge-Kutta Methods (RK4)
$y_{n+1}=y_n+\frac{h}{6}(k_1+2k_2+2k_3+k_4)$

where:

• $k_1=f(t_n,y_n)$
• $k_2=f(t_n+h/2,y_n+h{k}_1/2)$
• $k_3=f(t_n+h/2,y_n+h{k}_2/2)$
• $k_4=f(t_n+h,y_n+h{k}_3)$

Fourth-order accurate. Error proportional to $h^4$.

Stability: Implicit methods (backward Euler) more stable for stiff equations.

For PDEs

Finite Difference Methods
Approximate derivatives on a grid:
$\frac{\partial u}{\partial x}\approx \frac{u_{i+1}-u_{i-1}}{2\Delta x}$

Finite Element Methods (FEM)

• Divide domain into elements
• Approximate solution using basis functions
• Particularly good for irregular geometries

Applications

Equation	Application
$F=ma$ (Newton)	Classical mechanics
Heat equation	Thermal analysis, diffusion
Wave equation	Acoustics, electromagnetism, seismology
Schrödinger equation	Quantum mechanics
Navier-Stokes	Fluid dynamics
Maxwell’s equations	Electromagnetism
Black-Scholes	Option pricing in finance
Lotka-Volterra	Population dynamics
SIR model	Epidemiology

Tools

• Wolfram Alpha — symbolic solutions
• MATLAB — numerical computation and visualisation
• Python with SciPy — odeint, solve_ivp for ODEs
• Desmos — visualising slope fields
• Mathematica — symbolic and numerical methods

Learning Resources

Books

• Ordinary Differential Equations by Tenenbaum and Pollard — comprehensive classic
• Differential Equations by Blanchard, Devaney, Hall — modern approach with applications
• Partial Differential Equations: An Introduction by Walter Strauss — excellent first PDE text
• Fourier Series by Georgi Tolstov — rigorous treatment of Fourier analysis
• Applied Partial Differential Equations by Haberman — physics-oriented

Courses

• MIT 18.03 Differential Equations — Arthur Mattuck’s legendary lectures
• Khan Academy Differential Equations — accessible introduction
• 3Blue1Brown: Differential Equations — visual intuition

See Also

• Calculus
• Linear Algebra
• Complex Analysis