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Definition

Definitions

In what follows,

y

is a dependent variable representing an unknown function

y=f(x)

of the independent variable

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

{\frac {dy}{dx}},{\frac {d^{2}y}{dx^{2}}},\ldots ,{\frac {d^{n}y}{dx^{n}}}

is more useful for differentiation and integration, whereas Lagrange's notation

y',y'',\ldots ,y^{(n)}

is more useful for representing higher-order derivatives compactly, and Newton's notation

({\dot {y}},{\ddot {y}},{\overset {...}{y}})

is often used in physics for representing derivatives of low order with respect to time. Given

F

, a function of

x

,

y

, and derivatives of

y

. Then an equation of the form

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

is called an explicit ordinary differential equation of order

n

. More generally, an implicit ordinary differential equation of order

n

takes the form:

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

There are further classifications: Autonomous A differential equation is autonomous if it does not depend on the variable x.Linear A differential equation is linear if

F

can be written as a linear combination of the derivatives of

y

; that is, it can be rewritten as

y^{(n)}=\sum _{i=0}^{n-1}a_{i}(x)y^{(i)}+r(x)

where

a_{i}(x)

and

r(x)

are continuous functions of

x

. The function

r(x)

is called the source term, leading to further classification. Homogeneous A linear differential equation is homogeneous if

r(x)=0

. In this case, there is always the "trivial solution"

y=0

.Nonhomogeneous (or inhomogeneous) A linear differential equation is nonhomogeneous if

r(x)\neq 0

. Non-linear A differential equation that is not linear. A number of coupled differential equations form a system of equations. If

\mathbf {y}

is a vector whose elements are functions;

\mathbf {y} (x)=[y_{1}(x),y_{2}(x),\ldots ,y_{m}(x)]

, and

\mathbf {F}

is a vector-valued function of

\mathbf {y}

and its derivatives, then

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

is an explicit system of ordinary differential equations of order

n

and 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} '',\ldots ,\mathbf {y} ^{(n-1)}\right)\\f_{2}\left(x,\mathbf {y} ,\mathbf {y} ',\mathbf {y} '',\ldots ,\mathbf {y} ^{(n-1)}\right)\\\vdots \\f_{m}\left(x,\mathbf {y} ,\mathbf {y} ',\mathbf {y} '',\ldots ,\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} '',\ldots ,\mathbf {y} ^{(n)}\right)={\boldsymbol {0}}

where

{\boldsymbol {0}}=(0,0,\ldots ,0)

is the zero vector. In matrix form

{\begin{pmatrix}f_{1}(x,\mathbf {y} ,\mathbf {y} ',\mathbf {y} '',\ldots ,\mathbf {y} ^{(n)})\\f_{2}(x,\mathbf {y} ,\mathbf {y} ',\mathbf {y} '',\ldots ,\mathbf {y} ^{(n)})\\\vdots \\f_{m}(x,\mathbf {y} ,\mathbf {y} ',\mathbf {y} '',\ldots ,\mathbf {y} ^{(n)})\end{pmatrix}}={\begin{pmatrix}0\\0\\\vdots \\0\end{pmatrix}}

For a system of the form

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

, some sources also require that the Jacobian matrix

{\frac {\partial \mathbf {F} (x,\mathbf {u} ,\mathbf {v} )}{\partial \mathbf {v} }}

be non-singular in order to call this an implicit ODE [system]; an implicit ODE system satisfying this Jacobian non-singularity condition can be transformed into an explicit ODE system. In the same sources, implicit ODE systems with a singular Jacobian are termed differential algebraic equations (DAEs). This distinction is not merely one of terminology; DAEs have fundamentally different characteristics and are generally more involved to solve than (nonsingular) ODE systems. Presumably for additional derivatives, the Hessian matrix and so forth are also assumed non-singular according to this scheme, although note that any ODE of order greater than one can be (and usually is) rewritten as system of ODEs of first order, which makes the Jacobian singularity criterion sufficient for this taxonomy to be comprehensive at all orders. The behavior of a system of ODEs can be visualized through the use of a phase portrait. Given a differential equation

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

a function

u:I\subset \mathbb {R} \to \mathbb {R}

, where

I

is an interval, is called a solution or integral curve for

F

, if

u

is

n

-times differentiable on

I

, and

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

Given two solutions

u:J\subset \mathbb {R} \to \mathbb {R}

and

v:I\subset \mathbb {R} \to \mathbb {R}

,

u

is called an extension of

v

if

I\subset J

and

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

A solution that has no extension is called a maximal solution. A solution defined on all of

\mathbb {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'. A singular solution is a solution that cannot be obtained by assigning definite values to the arbitrary constants in the general solution. In the context of linear ODE, the terminology particular solution can also refer to any solution of the ODE (not necessarily satisfying the initial conditions), which is then added to the homogeneous solution (a general solution of the homogeneous ODE), which then forms a general solution of the original ODE. This is the terminology used in the guessing method section in this article, and is frequently used when discussing the method of undetermined coefficients and variation of parameters. For non-linear autonomous ODEs it is possible under some conditions to develop solutions of finite duration, meaning here that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after. These finite-duration solutions can't be analytical functions on the whole real line, and because they will be non-Lipschitz functions at their ending time, they are not included in the uniqueness theorem of solutions of Lipschitz differential equations. As example, the equation:

y'=-{\text{sgn}}(y){\sqrt {|y|}},\,\,y(0)=1

Admits the finite duration solution:

y(x)={\frac {1}{4}}\left(1-{\frac {x}{2}}+\left|1-{\frac {x}{2}}\right|\right)^{2}

- ^ ^{a}^{b}Harper (1976, p. 127) - ^Kreyszig (1972, p. 2) - ^Simmons (1972, p. 3) - ^ ^{a}^{b}Kreyszig (1972, p. 24) - ^Simmons (1972, p. 47) - ^Harper (1976, p. 128) - ^Kreyszig (1972, p. 12) - ^Ascher & Petzold (1998, p. 12) - ^Achim Ilchmann; Timo Reis (2014). Surveys in Differential-Algebraic Equations II. Springer. pp. 104–105. ISBN978-3-319-11050-9. - ^Ascher & Petzold (1998, p. 5) - ^Kreyszig (1972, p. 78) - ^Kreyszig (1972, p. 4) - ^Vardia T. Haimo (1985). "Finite Time Differential Equations". 1985 24th IEEE Conference on Decision and Control. pp. 1729–1733. doi:10.1109/CDC.1985.268832. S2CID45426376.

Definition

General definition

Given

F

, a function of

x

,

y

, and derivatives of

y

. Then an equation of the form

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

is called an explicit ordinary differential equation of order

n

. More generally, an implicit ordinary differential equation of order

n

takes the form:

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

There are further classifications: Autonomous A differential equation is autonomous if it does not depend on the variable x.Linear A differential equation is linear if

F

can be written as a linear combination of the derivatives of

y

; that is, it can be rewritten as

y^{(n)}=\sum _{i=0}^{n-1}a_{i}(x)y^{(i)}+r(x)

where

a_{i}(x)

and

r(x)

are continuous functions of

x

. The function

r(x)

is called the source term, leading to further classification. Homogeneous A linear differential equation is homogeneous if

r(x)=0

. In this case, there is always the "trivial solution"

y=0

.Nonhomogeneous (or inhomogeneous) A linear differential equation is nonhomogeneous if

r(x)\neq 0

. Non-linear A differential equation that is not linear. - ^ ^{a}^{b}Harper (1976, p. 127) - ^Kreyszig (1972, p. 2) - ^Simmons (1972, p. 3) - ^ ^{a}^{b}Kreyszig (1972, p. 24) - ^Simmons (1972, p. 47) - ^Harper (1976, p. 128)

Theorem

Local existence and uniqueness theorem simplified

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

y'=F(x,y)\,,\quad y_{0}=y(x_{0})

if

F

and

\partial F/\partial 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\in \mathbb {R}

) and

\times

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\in \mathbb {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 that take the form

F(x,y)

, and it can also be applied to systems of equations. - ^Elementary Differential Equations and Boundary Value Problems (4th Edition), W.E. Boyce, R.C. Diprima, Wiley International, John Wiley & Sons, 1986, ISBN0-471-83824-1

Theorem

Picard-Lindelöf Theorem

Intermediate

Guarantees the existence and uniqueness of solutions to first-order ODEs under certain continuity conditions.\n

y\'(t) = f(t, y(t)), \quad y(t_0) = y_0

Sequence of Expressions