Ordinary differential equation

Differential equations

Scope Natural sciences and engineering Astronomy Physics Chemistry Biology Geology Applied maths Continuum mechanics Chaos theory Dynamical systems Social sciences Economics Population dynamics
Classification
Operations Differential operator Notation for differentiation Ordinary Partial Differential-algebraic Integro-differential Fractional Linear Non-linear
Attributes of variables Dependent and independent variables Homogeneous Nonhomogeneous Coupled Decoupled Order Degree Autonomous Exact differential equation Complex differential equation
Relation to processes Difference (discrete analogue) stochastic Delay
Solutions
Solution topics Picard–Lindelöf theorem (existence and uniqueness) Wronskian Phase portrait Phase space Lyapunov stability Asymptotic stability Exponential stability Rate of convergence Series solutions Integral solutions Numerical integration Dirac Delta function
Solution methods Inspection Separation of variables Method of undetermined coefficients Integrating factor Integral transforms Euler method Finite difference method Crank–Nicolson method Runge–Kutta methods Finite element analysis Galerkin method Perturbation theory
Mathematicians Isaac Newton Leonhard Euler Charles Émile Picard Józef Maria Hoene-Wroński Ernst Lindelöf Rudolf Lipschitz Augustin-Louis Cauchy John Crank Phyllis Nicolson Carl David Tolmé Runge Martin Wilhelm Kutta

See also: Examples of differential equations

In mathematics, an ordinary differential equation (abbreviated ODE) is an equation containing a function of one independent variable and its derivatives. There are many general forms an ODE can take, and these are classified in practice (see below).^[1]^[2] The derivatives are ordinary because partial derivatives only apply to functions of many independent variables (see Partial differential equation).

The subject of ODEs is a sophisticated one (more so with PDEs), primarily due to the various forms the ODE can take and how they can be integrated. Linear differential equations are ones with solutions that can be added and multiplied by coefficients, and the theory of linear differential equations is well-defined and understood, and exact closed form solutions can be obtained. By contrast, ODEs which do not have additive solutions are non-linear, and finding the solutions is much more sophisticated because it is rarely possible to represent them by elementary functions in closed form — rather the exact (or "analytic") solutions are in series or integral form. Frequently graphical and numerical methods are used to generate solutions, by hand or on computer (only approximately, but possible to do very accurately depending on the specific method used), because in this way the properties of the solutions without solving them can still yield very useful information, which may be all that is needed.

Background

The trajectory of a projectile launched from a cannon follows a curve determined by an ordinary differential equation that is derived from Newton's second law.

Ordinary differential equations arise in many different contexts throughout mathematics and science (social and natural) one way or another, because when describing changes mathematically, the most accurate way uses differentials and derivatives (related, though not quite the same). Since various differentials, derivatives, and functions become inevitably related to each other via equations, a differential equation is the result, governing dynamical phenomena, evolution and variation. Often, quantities are defined as the rate of change of other quantities (time derivatives), or gradients of quantities, which is how they enter differential equations.

Specific mathematical fields include geometry and analytical mechanics. Scientific fields include much of physics and astronomy (celestial mechanics), geology (weather modelling), chemistry (reaction rates),^[3] biology (infectious diseases, genetic variation), ecology and population modelling (population competition), economics (stock trends, interest rates and the market equilibrium price changes).

Many mathematicians have studied differential equations and contributed to the field, including Newton, Leibniz, the Bernoulli family, Riccati, Clairaut, d'Alembert and Euler.

A simple example is Newton's second law of motion — the relationship between the displacement x and the time t of the object under the force F which leads to the differential equation

$m \frac{\mathrm{d}^2 x(t)}{\mathrm{d}t^2} = F(x(t)),\,$

for the motion of a particle of constant mass m. In general, F depends on the position x(t) of the particle at time t, and so the unknown function x(t) appears on both sides of the differential equation, as is indicated in the notation F(x(t)).^[4]^[5]^[6]^[7]

Definitions

In what follows, let y be a dependent variable and x an independent variable, so that y = y(x) is an unknown function in x. The notation for differentiation varies depending upon the author and upon which notation is most useful for the task at hand. In this context the Leibniz's notation (dy/dx,d²y/dx²,...dⁿy/dxⁿ) is useful for differentials and when integration is to be done, while Newton's and Lagrange's notation (y′,y′′, ... y⁽ⁿ⁾) is useful for representing derivatives of any order compactly.

General definition of an ODE

Let F be a given function of x, y, and derivatives of y. Then an equation of the form

$F\left (x,y,y',\cdots y^{(n-1)} \right )=y^{(n)}$

is called an explicit ordinary differential equation of order n.^[8]^[9]

More generally, an implicit ordinary differential equation of order n takes the form:^[10]

$F\left(x, y, y', y'',\ \cdots,\ y^{(n)}\right) = 0$

There are further classifications:

Autonomous

A differential equation not depending on x is called autonomous.

Linear

A differential equation is said to be linear if F can be written as a linear combination of the derivatives of y:

$y^{(n)} = \sum_{i=0}^{n-1} a_i(x) y^{(i)} + r(x)$

where a_i(x) and r(x) continuous functions in x.^[11]^[12]^[13] Non-linear equations cannot be written in this form. The function r(x) is called the source term, leading to two further important classifications:^[14]^[15]

Homogeneous: If r(x) = 0, and consequently one "automatic" solution is the trivial solution, y = 0. The solution of a linear homogeneous equation is a complementary function, denoted here by y_c.

Nonhomogeneous (or inhomogeneous): If r(x) ≠ 0. The additional solution to the complementary function is the particular integral, denoted here by y_p.

The general solution to a linear equation can be written as y = y_c + y_p.

System of ODEs

A number of coupled differential equations form a system of equations. If y is a vector whose elements are functions; y(x) = [y₁(x), y₂(x),..., y_m(x)], and F is a vector valued function of y and its derivatives, then

$\mathbf{y}^{(n)} = \mathbf{F}\left(x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\cdots \mathbf{y}^{(n-1)} \right)$

is an explicit system of ordinary differential equations of order or dimension m. In column vector form:

$\begin{pmatrix}y_1^{(n)} \y_2^{(n)} \\vdots \y_m^{(n)}\end{pmatrix} = \begin{pmatrix}F_1 \left (x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\cdots \mathbf{y}^{(n-1)} \right ) \F_2 \left (x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\cdots \mathbf{y}^{(n-1)} \right ) \\vdots \F_m \left (x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\cdots \mathbf{y}^{(n-1)} \right) \\end{pmatrix}$

These are not necessarily linear. The implicit analogue is:

$\mathbf{F} \left(x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\cdots \mathbf{y}^{(n)} \right) = \boldsymbol{0}$

where 0 = (0, 0,... 0) is the zero vector. In matrix form

$\begin{pmatrix}F_1(x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\cdots \mathbf{y}^{(n)}) \F_2(x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\cdots \mathbf{y}^{(n)}) \\vdots \F_m(x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\cdots \mathbf{y}^{(n)}) \\end{pmatrix}=\begin{pmatrix}0\0\\vdots\0\\end{pmatrix}$

Solutions

Given a differential equation

$F\left(x, y, y', \cdots, y^{(n)} \right) = 0$

a function u: I ⊂ R → R is called the solution or integral curve for F, if u is n-times differentiable on I, and

$F(x,u,u',\ \cdots,\ u^{(n)})=0 \quad x \in I.$

Given two solutions u: J ⊂ R → R and v: I ⊂ R → R, u is called an extension of v if I ⊂ J and

$u(x) = v(x) \quad x \in I.\,$

A solution which has no extension is called a maximal solution. A solution defined on all of R is called a global solution.

A general solution of an n-th order equation is a solution containing n arbitrary independent constants of integration. A particular solution is derived from the general solution by setting the constants to particular values, often chosen to fulfill set 'initial conditions or boundary conditions'.^[16] A singular solution is a solution which cannot be obtained by assigning definite values to the arbitrary constants in the general solution.^[17]

Theories of ODEs

Singular solutions

The theory of singular solutions of ordinary and partial differential equations was a subject of research from the time of Leibniz, but only since the middle of the nineteenth century did it receive special attention. A valuable but little-known work on the subject is that of Houtain (1854). Darboux (starting in 1873) was a leader in the theory, and in the geometric interpretation of these solutions he opened a field which was worked by various writers, notably Casorati and Cayley. To the latter is due (1872) the theory of singular solutions of differential equations of the first order as accepted circa 1900.

Reduction to quadratures

The primitive attempt in dealing with differential equations had in view a reduction to quadratures. As it had been the hope of eighteenth-century algebraists to find a method for solving the general equation of the nth degree, so it was the hope of analysts to find a general method for integrating any differential equation. Gauss (1799) showed, however, that the differential equation meets its limitations very soon unless complex numbers are introduced. Hence analysts began to substitute the study of functions, thus opening a new and fertile field. Cauchy was the first to appreciate the importance of this view. Thereafter the real question was to be, not whether a solution is possible by means of known functions or their integrals, but whether a given differential equation suffices for the definition of a function of the independent variable or variables, and if so, what are the characteristic properties of this function.

Fuchsian theory

Main article: Frobenius method

Two memoirs by Fuchs (Crelle, 1866, 1868), inspired a novel approach, subsequently elaborated by Thomé and Frobenius. Collet was a prominent contributor beginning in 1869, although his method for integrating a non-linear system was communicated to Bertrand in 1868. Clebsch (1873) attacked the theory along lines parallel to those followed in his theory of Abelian integrals. As the latter can be classified according to the properties of the fundamental curve which remains unchanged under a rational transformation, so Clebsch proposed to classify the transcendent functions defined by the differential equations according to the invariant properties of the corresponding surfaces f = 0 under rational one-to-one transformations.

Lie's theory

From 1870 Sophus Lie's work put the theory of differential equations on a more satisfactory foundation. He showed that the integration theories of the older mathematicians can, by the introduction of what are now called Lie groups, be referred to a common source; and that ordinary differential equations which admit the same infinitesimal transformations present comparable difficulties of integration. He also emphasized the subject of transformations of contact.

Lie's group theory of differential equations, has been certified, namely: (1) that it unifies the many ad hoc methods known for solving differential equations, and (2) that it provides powerful new ways to find solutions. The theory has applications to both ordinary and partial differential equations.^[18]

A general approach to solve DE's uses the symmetry property of differential equations, the continuous infinitesimal transformations of solutions to solutions (Lie theory). Continuous group theory, Lie algebras and differential geometry are used to understand the structure of linear and nonlinear (partial) differential equations for generating integrable equations, to find its Lax pairs, recursion operators, Bäcklund transform and finally finding exact analytic solutions to the DE.

Symmetry methods have been recognized to study differential equations arising in mathematics, physics, engineering, and many other disciplines.

Sturm–Liouville theory

Main article: Sturm–Liouville theory

Sturm–Liouville theory is a theory of eigenvalues and eigenfunctions of linear operators defined in terms of second-order homogeneous linear equations, and is useful in the analysis of certain partial differential equations.

Existence and uniqueness of solutions

There are several theorems that establish existence and uniqueness of solutions to initial value problems involving ODEs both locally and globally. The two main theorems are

Theorem	Assumption	Conclusion
Peano existence theorem	F continuous	local existence only
Picard–Lindelöf theorem	F Lipschitz continuous	local existence and uniqueness

which are both local results.

Local existence and uniqueness theorem simplified

The theorem can be stated simply as follows.^[19] For the equation and initial value problem:

$y' = F(x,y)\,,\quad y_0 = y(x_0)$

if F and ∂F/∂y are continuous in a closed rectangle

$R=[x_0-a,x_0+a]\times [y_0-b,y_0+b]$

in the x-y plane, where a and b are real (symbolically: a, b ∈ ℝ) and × denotes the cartesian product, square brackets denote closed intervals, then there is an interval

$I = [x_0-h,x_0+h] \subset [x_0-a,x_0+a]$

for some h ∈ ℝ where the solution to the above equation and initial value problem can be found. That is, there is a solution and it is unique. Since there is no restriction on F to be linear, this applies to non-linear equations which take the form F(x, y), and it can also be applied to systems of equations.

Global uniqueness and maximum domain of solution

When the hypotheses of the Picard–Lindelöf theorem are satisfied, then local existence and uniqueness can be extended to a global result. More precisely:^[20]

For each initial condition (x₀, y₀) there exists a unique maximum (possibly infinite) open interval

$I_{max} = (x_-,x_+), x_\pm \in \mathbb{R}, x_0 \in I_{max}$

such that any solution which satisfies this initial condition is a restriction of the solution which satisfies this initial condition with domain I_max.

In the case that $x_\pm \nrightarrow \pm\infty$ , there are exactly two possibilities

explosion in finite time: $\lim_{x \to x_\pm} \|y(x)\| \rightarrow \infty$
leaves domain of definition: $\lim_{x \to x_\pm} \in \partial \bar{\Omega}$

where Ω is the open set in which F is defined, and $\partial \bar{\Omega}$ is its boundary.

Note that the maximum domain of the solution

is always an interval (to have uniqueness)
may be smaller than ℝ
may depend on the specific choice of (x₀, y₀).

Example

$y' = y^2$

This means that F(x, y) = y², which is C¹ and therefore Lipschitz continuous for all y, satisfying the Picard–Lindelöf theorem.

Even in such a simple setting, the maximum domain of solution cannot be all ℝ, since the solution is

$y(x) = \frac{y_0}{(x_0-x)y_0+1}$

which has maximum domain:

$\begin{cases} \mathbb{R} & y_0 = 0 \ (-\infty, x_0+\frac{1}{y_0}) & y_0 > 0 \ (x_0+\frac{1}{y_0},+\infty) & y_0 < 0 \end{cases}$

This shows clearly that the maximum interval may depend on the initial conditions. The domain of y could be taken as being $\mathbf{R} \smallsetminus (x_0+ 1/y_0)$ , but this would lead to a domain that is not an interval, so that the side opposite to the initial condition would be disconnected from the initial condition, and therefore not uniquely determined by it.

The maximum domain is not ℝ because

$\lim_{x \to x_\pm} \|y(x)\| \rightarrow \infty\,,$

which is one of the two possible cases according to the above theorem.

Reduction of order

Differential equations can usually be solved more easily if the order of the equation can be reduced.

Reduction to a first order system

Any differential equation of order n,

$F\left(x, y, y', y'',\ \cdots,\ y^{(n-1)}\right) = y^{(n)}$

can be written as a system of n first-order differential equations by defining a new family of unknown functions

$y_i = y^{(i-1)}.\!$

for i = 1, 2,... n. The n-dimensional system of first-order coupled differential equations is then

$\begin{array}{rcl} y_1'&=&y_2\ y_2'&=&y_3\ &\vdots&\ y_{n-1}'&=&y_n\ y_n'&=&F(x,y_1,\cdots,y_n).\end{array}$

more compactly in vector notation:

$\mathbf{y}'=\mathbf{F}(x,\mathbf{y})$

where

$\mathbf{y}=(y_1,\cdots,y_n),\quad \mathbf{F}(x,y_1,\cdots,y_n)=(y_2,\cdots,y_n,F(x,y_1,\cdots,y_n)).$

Summary of exact solutions

Some differential equations have solutions which can be written in an exact and closed form. Several important classes are given here.

In the table below, P(x), Q(x), P(y), Q(y), and M(x,y), N(x,y) are any integrable functions of x, y, and b and c are real given constants, and C₁, C₂,... are arbitrary constants (complex in general). The differential equations are in their equivalent and alternative forms which lead to the solution through integration.

In the integral solutions, λ and ε are dummy variables of integration (the continuum analogues of indices in summation), and the notation ∫^xF(λ)dλ just means to integrate F(λ) with respect to λ, then after the integration substitute λ = x, without adding constants (explicitly stated).

Differential equation	Solution method	General solution
Separable equations
First order, separable in x and y (general case, see below for special cases)^[21] $P_1(x)Q_1(y) + P_2(x)Q_2(y)\,\frac{dy}{dx} = 0 \,\!$ $P_1(x)Q_1(y)\,dx + P_2(x)Q_2(y)\,dy = 0 \,\!$	Separation of variables (divide by P₂Q₁).	$\int^x \frac{P_1(\lambda)}{P_2(\lambda)}\,d\lambda + \int^y \frac{Q_2(\lambda)}{Q_1(\lambda)}\,d\lambda = C \,\!$
First order, separable in x^[22] $\frac{dy}{dx} = F(x)\,\!$ $dy= F(x) \, dx\,\!$	Direct integration.	$y= \int^x F(\lambda) \, d\lambda + C \,\!$
First order, autonomous, separable in y^[23] $\frac{dy}{dx} = F(y)\,\!$ $dy= F(y) \, dx\,\!$	Separation of variables (divide by F).	$x=\int^y \frac{d\lambda}{F(\lambda)}+C\,\!$
First order, separable in x and y^[24] $P(y)\frac{dy}{dx} + Q(x)= 0\,\!$ $P(y)\,dy + Q(x)\,dx =0\,\!$	Integrate throughout.	$\int^y P(\lambda)\,{d\lambda} + \int^x Q(\lambda)\,d\lambda = C\,\!$
General first order equations
First order, homogeneous^[25] $\frac{dy}{dx} = F \left( \frac{y}{x} \right ) \,\!$	Set y = ux, then solve by separation of variables in u and x.	$\ln (Cx) = \int^{y/x} \frac{d\lambda}{F(\lambda) - \lambda} \, \!$
First order, separable^[26] $yM(xy) + xN(xy)\,\frac{dy}{dx} = 0 \,\!$ $yM(xy)\,dx + xN(xy)\,dy = 0 \,\!$	Separation of variables (divide by xy).	$\ln (Cx) = \int^{xy} \frac{N(\lambda)\,d\lambda}{\lambda [N(\lambda)-M(\lambda)] } \,\!$ If N = M, the solution is xy = C.
Exact differential, first order^[27] $M(x,y) \frac{dy}{dx} + N(x,y) = 0 \,\!$ $M(x,y)\,dy + N(x,y)\,dx = 0 \,\!$ where $\frac{\partial M}{\partial x} = \frac{\partial N}{\partial y} \, \!$	Integrate throughout.	$\begin{align}F(x,y) & = \int^y M(x,\lambda)\,d\lambda + \int^x N(\lambda,y)\,d\lambda \ & + Y(y) + X(x) = C \end{align} \,\!$ where Y(y) and X(x) are functions from the integrals rather than constant values, which are set to make the final function F(x, y) satisfy the initial equation.
Inexact differential, first order^[28] $M(x,y) \frac{dy}{dx} + N(x,y) = 0 \,\!$ $M(x,y)\,dy + N(x,y)\,dx = 0 \,\!$ where $\frac{\partial M}{\partial x} \neq \frac{\partial N}{\partial y} \, \!$	Integration factor μ(x, y) satisfying $\frac{\partial (\mu M)}{\partial x} = \frac{\partial (\mu N)}{\partial y} \, \!$	If μ(x, y) can be found: $\begin{align}F(x,y) & = \int^y \mu(x,\lambda)M(x,\lambda)\,d\lambda + \int^x \mu(\lambda,y)N(\lambda,y)\,d\lambda \& + Y(y) + X(x) = C \\end{align} \, \!$
General second order equations
Second order, autonomous^[29] $\frac{d^2y}{dx^2} = F(y) \,\!$	Multiply equation by 2dy/dx, substitute $2 \frac{dy}{dx}\frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right)^2 \,\!$ , then integrate twice.	$x = \pm \int^y \frac{ d \lambda}{\sqrt{2 \int^\lambda F(\epsilon) \, d \epsilon + C_1}} + C_2 \, \!$
Linear equations (up to nth order)
First order, linear, inhomogeneous, function coefficients^[30] $\frac{dy}{dx} + P(x)y=Q(x)\,\!$	Integrating factor: $e^{\int^x P(\lambda)\,d\lambda}$ .	$y = e^{- \int^x P(\lambda) \, d\lambda}\left[\int^x e^{\int^\lambda P(\epsilon) \, d\epsilon}Q(\lambda) \, {d\lambda} +C \right]$
Second order, linear, inhomogeneous, constant coefficients^[31] $\frac{d^2y}{dx^2} + b\frac{dy}{dx} + cy = r(x)\,\!$	Complementary function y_c: assume y_c = e^αx, substitute and solve polynomial in α, to find the linearly independent functions $e^{\alpha_j x}$ . Particular integral y_p: in general the method of variation of parameters, though for very simple r(x) inspection may work.^[32]	$y=y_c+y_p$ If b² > 4c, then: $y_c=C_1e^{ \left ( -b+\sqrt{b^2 - 4c} \right )\frac{x}{2}} + C_2e^{-\left ( b+\sqrt{b^2 - 4c} \right )\frac{x}{2}}\,\!$ If b² = 4c, then: $y_c = (C_1x + C_2)e^{-bx/2}\,\!$ If b² < 4c, then: $y_c = e^{ -b\frac{x}{2}} \left [ C_1 \sin{\left ( \sqrt{\left \| b^2-4c \right \|}\frac{x}{2} \right )} + C_2\cos{\left ( \sqrt{\left \| b^2-4c \right \|}\frac{x}{2} \right )} \right ] \,\!$
nth order, linear, inhomogeneous, constant coefficients^[33] $\sum_{j=0}^n b_j \frac{d^j y}{dx^j} = r(x)\,\!$	Complementary function y_c: assume y_c = e^αx, substitute and solve polynomial in α, to find the linearly independent functions $e^{\alpha_j x}$ . Particular integral y_p: in general the method of variation of parameters, though for very simple r(x) inspection may work.^[34]	$y=y_c+y_p$ Since α_j are the solutions of the polynomial of degree n: $\prod_{j=1}^n \left ( \alpha - \alpha_j \right ) = 0 \,\!$ , then: for α_j all different, $y_c = \sum_{j=1}^n C_j e^{\alpha_j x} \,\!$ for each root α_j repeated k_j times, $y_c = \sum_{j=1}^n \left( \sum_{\ell=1}^{k_j} C_\ell x^{\ell-1}\right )e^{\alpha_j x} \,\!$ for some α_j complex, then setting α = χ_j + iγ_j, and using Euler's formula, allows some terms in the previous results to be written in the form $C_je^{\alpha_j x} = C_j e^{\chi_j x}\cos(\gamma_j x + \phi_j)\,\!$ where ϕ_j is an arbitrary constant (phase shift).

Software for ODE solving

FuncDesigner (free license: BSD, uses Automatic differentiation, also can be used online via Sage-server)
Maxima computer algebra system (GPL)
COPASI a free (Artistic License 2.0) software package for the integration and analysis of ODEs.
MATLAB a Technical Computing Software (MATrix LABoratory)
GNU Octave a high-level language, primarily intended for numerical computations.
SciLab an open source software for numerical computation.
Maple
Mathematica

Notes

^ Kreyszig (1972, p. 1)
^ Simmons (1972, p. 2)
^ Mathematics for Chemists, D.M. Hirst, THE MACMILLAN PRESS, 1976, (No ISBN) SBN: 333-18172-7
^ Kreyszig (1972, p. 64)
^ Simmons (1972, pp. 1,2)
^ Halliday & Resnick (1977, p. 78)
^ Tipler (1991, pp. 78–83)
^ Harper (1976, p. 127)
^ Kreyszig (1972, p. 2)
^ Simmons (1972, p. 3)
^ Harper (1976, p. 127)
^ Kreyszig (1972, p. 24)
^ Simmons (1972, p. 47)
^ Harper (1976, p. 128)
^ Kreyszig (1972, p. 24)
^ Kreyszig (1972, p. 78)
^ Kreyszig (1972, p. 4)
^ Lawrence (1999, p. 9)
^ Elementary Differential Equations and Boundary Value Problems (4th Edition), W.E. Boyce, R.C. Diprima, Wiley International, John Wiley & Sons, 1986, ISBN 0-471-83824-1
^ Boscain; Chitour 2011, p. 21
^ Mathematical Handbook of Formulas and Tables (3rd edition), S. Lipschutz, M.R. Spiegel, J. Liu, Schuam's Outline Series, 2009, ISC_2N 978-0-07-154855-7
^ Elementary Differential Equations and Boundary Value Problems (4th Edition), W.E. Boyce, R.C. Diprima, Wiley International, John Wiley & Sons, 1986, ISC_2N 0-471-83824-1
^ Elementary Differential Equations and Boundary Value Problems (4th Edition), W.E. Boyce, R.C. Diprima, Wiley International, John Wiley & Sons, 1986, ISC_2N 0-471-83824-1
^ Elementary Differential Equations and Boundary Value Problems (4th Edition), W.E. Boyce, R.C. Diprima, Wiley International, John Wiley & Sons, 1986, ISC_2N 0-471-83824-1
^ Elementary Differential Equations and Boundary Value Problems (4th Edition), W.E. Boyce, R.C. Diprima, Wiley International, John Wiley & Sons, 1986, ISC_2N 0-471-83824-1
^ Mathematical Handbook of Formulas and Tables (3rd edition), S. Lipschutz, M.R. Spiegel, J. Liu, Schuam's Outline Series, 2009, ISC_2N 978-0-07-154855-7
^ Elementary Differential Equations and Boundary Value Problems (4th Edition), W.E. Boyce, R.C. Diprima, Wiley International, John Wiley & Sons, 1986, ISC_2N 0-471-83824-1
^ Elementary Differential Equations and Boundary Value Problems (4th Edition), W.E. Boyce, R.C. Diprima, Wiley International, John Wiley & Sons, 1986, ISC_2N 0-471-83824-1
^ Further Elementary Analysis, R. Porter, G.Bell & Sons (London), 1978, ISBN 0-7135-1594-5
^ Elementary Differential Equations and Boundary Value Problems (4th Edition), W.E. Boyce, R.C. Diprima, Wiley International, John Wiley & Sons, 1986, ISC_2N 0-471-83824-1
^ Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press, 2010, ISC_2N 978-0-521-86153-3
^ Elementary Differential Equations and Boundary Value Problems (4th Edition), W.E. Boyce, R.C. Diprima, Wiley International, John Wiley & Sons, 1986, ISC_2N 0-471-83824-1
^ Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press, 2010, ISC_2N 978-0-521-86153-3
^ Elementary Differential Equations and Boundary Value Problems (4th Edition), W.E. Boyce, R.C. Diprima, Wiley International, John Wiley & Sons, 1986, ISC_2N 0-471-83824-1

References

Halliday, David; Resnick, Robert (1977), Physics (3rd ed.), New York: Wiley, ISBN 0-471-71716-9
Harper, Charlie (1976), Introduction to Mathematical Physics, New Jersey: Prentice-Hall, ISBN 0-13-487538-9
Kreyszig, Erwin (1972), Advanced Engineering Mathematics (3rd ed.), New York: Wiley, ISBN 0-471-50728-8.
Polyanin, A. D. and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations (2nd edition)", Chapman & Hall/CRC Press, Boca Raton, 2003. ISBN 1-58488-297-2
Simmons, George F. (1972), Differential Equations with Applications and Historical Notes, New York: McGraw-Hill, LCCN 75-173716
Tipler, Paul A. (1991), Physics for Scientists and Engineers: Extended version (3rd ed.), New York: Worth Publishers, ISBN 0-87901-432-6
Boscain, Ugo; Chitour, Yacine (2011) (in french), Introduction à l'automatique, http://www.cmapx.polytechnique.fr/~bo scain/poly2011.pdf
Lawrence, Dresner (1999), Applications of Lie's Theory of Ordinary and Partial Differential Equations, Bristol and Philadelphia: Institute of Physics Publishing

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Ibragimov, Nail H (1993). CRC Handbook of Lie Group Analysis of Differential Equations Vol. 1-3. Providence: CRC-Press. ISBN 0-8493-4488-3 .
Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0.
A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, Handbook of First Order Partial Differential Equations, Taylor & Francis, London, 2002. ISBN 0-415-27267-X
D. Zwillinger, Handbook of Differential Equations (3rd edition), Academic Press, Boston, 1997.

External links

Hazewinkel, Michiel, ed. (2001), "Differential equation, ordinary", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4, http://www.encyclopediaofmath.org/ind ex.php?title=p/d031910

Differential Equations at the Open Directory Project (includes a list of software for solving differential equations).
EqWorld: The World of Mathematical Equations, containing a list of ordinary differential equations with their solutions.
Online Notes / Differential Equations by Paul Dawkins, Lamar University.
Differential Equations, S.O.S. Mathematics.
A primer on analytical solution of differential equations from the Holistic Numerical Methods Institute, University of South Florida.
Ordinary Differential Equations and Dynamical Systems lecture notes by Gerald Teschl.
Notes on Diffy Qs: Differential Equations for Engineers An introductory textbook on differential equations by Jiri Lebl of UIUC.

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