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Since Newton, mankind has come to realize that the laws of physics are always expressed in the language of differential equations. - Steven Strogatz

Differential equations relate functions and their derivatives, and are pervasive in modern engineering and science. The study of differential equations is vast with many innovative ideas being explored.

An incomplete classification of equations

ClassificationTypeDescriptionExample
Order of Differential EquationsFirst-OrderInvolves the first derivative of the function.dydx=y\frac{\mathrm{d}y}{\mathrm{d}x} = y
Second-Order and HigherInvolves second or higher derivatives.d2ydx2+3dydx+2y=0\frac{\mathrm{d}^2y}{\mathrm{d}x^2} + 3\frac{\mathrm{d}y}{\mathrm{d}x} + 2y = 0
Linear vs. Nonlinear Differential EquationsLinearThe dependent variable and its derivatives appear linearly.d2ydx2+p(x)dydx+q(x)y=g(x)\frac{\mathrm{d}^2y}{\mathrm{d}x^2} + p(x)\frac{\mathrm{d}y}{\mathrm{d}x} + q(x)y = g(x)
NonlinearThe equation involves nonlinear terms of the dependent variable or its derivatives.(dydx)2+y=0\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2 + y = 0
Homogeneous vs. Non-Homogeneous Differential EquationsHomogeneousAll terms are a function of the dependent variable and its derivatives.d2ydx2−y=0\frac{\mathrm{d}^2y}{\mathrm{d}x^2} - y = 0
Non-HomogeneousIncludes terms that are not a function of the dependent variable or its derivatives.d2ydx2−y=ex\frac{\mathrm{d}^2y}{\mathrm{d}x^2} - y = e^x
Initial / Boundary Value ProblemsInitial Value ProblemsThe solution is determined by the value of the function and its derivatives at a single point.dydx=y,y(0)=1\frac{\mathrm{d}y}{\mathrm{d}x} = y, \quad y(0) = 1
Boundary Value ProblemsThe solution is determined by the values of the function at multiple points.d2ydx2=−y,y(0)=0,y(π)=0\frac{\mathrm{d}^2y}{\mathrm{d}x^2} = -y, \quad y(0) = 0, \quad y(\pi) = 0
Ordinary vs. Partial Differential EquationsOrdinary Differential Equations (ODEs)Involve functions of a single variable and their derivatives.dydx+y=0\frac{\mathrm{d}y}{\mathrm{d}x} + y = 0
Partial Differential Equations (PDEs)Involve functions of multiple variables and their partial derivatives.∂u∂t=c2∂2u∂x2\frac{\partial u}{\partial t} = c^2 \frac{\partial^2 u}{\partial x^2}
Time-Dependent PDEsElliptic PDEsStationary problems in time.Laplace’s equation, ∇2u=0\nabla^2 u = 0
Parabolic PDEsFirst derivative in time.The heat equation, ∂u∂t=α∇2u\frac{\partial u}{\partial t} = \alpha \nabla^2 u
Hyperbolic PDEsSecond derivative in time.The wave equation, ∂2u∂t2=c2∇2u\frac{\partial^2 u}{\partial t^2} = c^2 \nabla^2 u
Executable Engineering
Gaussian quadrature
Executable Engineering
Initial value problems