Ordinary Differential Equations
First-order equations, separable models, linear ODEs, equilibrium solutions and second-order equations.
Learn ODEs →MATHEMATICS · CHANGE THROUGH LAWS
Differential Equations
Differential equations describe how a quantity changes rather than only what its value is. They are the mathematical language behind growth, motion, circuits, diffusion, waves, populations and control systems.
What is a differential equation?
A differential equation relates an unknown function to one or more of its derivatives. A first-order ordinary differential equation (ODE), for example, can be written as
\[y'=f(x,y).\]
The equation tells us the local rate of change. A solution is a function whose derivatives satisfy the equation on a stated interval. An initial condition such as \(y(x_0)=y_0\) selects a particular solution from a family.
Core idea: solving a differential equation means recovering a function from information about how it changes.
Recommended learning path
Systems of ODEs
Vector form, phase portraits, eigenvalues, stability and coupled dynamical systems.
Explore ODE systems →PDE Classification
Elliptic, parabolic and hyperbolic partial differential equations and what the classification means.
Classify PDEs →PDE Methods
Separation of variables, Fourier methods and boundary or initial conditions.
Study PDE methods →Standard PDEs
Heat, wave and Laplace equations as foundational models of diffusion, propagation and equilibrium.
See standard PDEs →Transforms & Special Functions
Laplace/Fourier transforms plus functions such as Bessel and Legendre functions that arise naturally in differential equations.
Explore transforms →How to read a differential equation
1
Identify the unknown.
Is the unknown a scalar function \(y(x)\), a vector \(\mathbf{x}(t)\), or a field \(u(x,t)\)?
2
Determine the order.
The highest derivative present determines the order. For example, \(y''+4y=0\) is second order.
3
Check linearity.
A linear ODE has the unknown function and its derivatives only to the first power and not multiplied together.
4
Include the data.
Initial or boundary conditions are part of the mathematical problem, not an optional afterthought.
Three model equations worth recognizing
Exponential growth/decay
\[y'=ky\]
Solution: \(y=Ce^{kx}\). This appears in population growth, radioactive decay and continuous compounding.
Harmonic oscillator
\[y''+\omega^2y=0\]
Solutions are sinusoidal and model ideal springs, oscillations and wave modes.
Heat equation
\[u_t=\alpha u_{xx}\]
A canonical parabolic PDE describing diffusion and smoothing over time.
Wave equation
\[u_{tt}=c^2u_{xx}\]
A canonical hyperbolic PDE describing propagation at finite speed.
Connection to Calculus and Linear Algebra
Differential equations sit at the intersection of several subjects. Derivatives provide the local change laws, integrals recover accumulated change, and linear algebra becomes essential for coupled systems and stability.