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Definitions

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The statement of the theorem

In what follows, yy is a dependent variable representing an unknown function y=f(x)y=f(x) of the independent variable xx . 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 dydx,d2ydx2,…,dnydxn{\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′′,…,y(n)y',y'',\ldots ,y^{(n)} is more useful for representing higher-order derivatives compactly, and Newton's notation (y˙,y¨,y...)({\dot {y}},{\ddot {y}},{\overset {...}{y}}) is often used in physics for representing derivatives of low order with respect to time. Given FF , a function of xx , yy , and derivatives of yy . Then an equation of the form F(x,y,y′,…,y(n−1))=y(n)F\left(x,y,y',\ldots ,y^{(n-1)}\right)=y^{(n)} is called an explicit ordinary differential equation of order nn . More generally, an implicit ordinary differential equation of order nn takes the form: F(x,y,y′,y′′, …, y(n))=0F\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 FF can be written as a linear combination of the derivatives of yy ; that is, it can be rewritten as y(n)=∑i=0n−1ai(x)y(i)+r(x)y^{(n)}=\sum _{i=0}^{n-1}a_{i}(x)y^{(i)}+r(x) where ai(x)a_{i}(x) and r(x)r(x) are continuous functions of xx . The function r(x)r(x) is called the source term, leading to further classification. Homogeneous A linear differential equation is homogeneous if r(x)=0r(x)=0 . In this case, there is always the "trivial solution" y=0y=0 .Nonhomogeneous (or inhomogeneous) A linear differential equation is nonhomogeneous if r(x)≠0r(x)\neq 0 . Non-linear A differential equation that is not linear. A number of coupled differential equations form a system of equations. If y\mathbf {y} is a vector whose elements are functions; y(x)=[y1(x),y2(x),…,ym(x)]\mathbf {y} (x)=[y_{1}(x),y_{2}(x),\ldots ,y_{m}(x)] , and F\mathbf {F} is a vector-valued function of y\mathbf {y} and its derivatives, then y(n)=F(x,y,y′,y′′,…,y(n−1))\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 nn and dimension mm . In column vector form: (y1(n)y2(n)⋮ym(n))=(f1(x,y,y′,y′′,…,y(n−1))f2(x,y,y′,y′′,…,y(n−1))⋮fm(x,y,y′,y′′,…,y(n−1))){\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: F(x,y,y′,y′′,…,y(n))=0\mathbf {F} \left(x,\mathbf {y} ,\mathbf {y} ',\mathbf {y} '',\ldots ,\mathbf {y} ^{(n)}\right)={\boldsymbol {0}} where 0=(0,0,…,0){\boldsymbol {0}}=(0,0,\ldots ,0) is the zero vector. In matrix form (f1(x,y,y′,y′′,…,y(n))f2(x,y,y′,y′′,…,y(n))⋮fm(x,y,y′,y′′,…,y(n)))=(00⋮0){\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 F(x,y,y′)=0\mathbf {F} \left(x,\mathbf {y} ,\mathbf {y} '\right)={\boldsymbol {0}} , some sources also require that the Jacobian matrix ∂F(x,u,v)∂v{\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(x,y,y′,…,y(n))=0F\left(x,y,y',\ldots ,y^{(n)}\right)=0 a function u:I⊂R→Ru:I\subset \mathbb {R} \to \mathbb {R} , where II is an interval, is called a solution or integral curve for FF , if uu is nn -times differentiable on II , and F(x,u,u′, …, u(n))=0x∈I.F(x,u,u',\ \ldots ,\ u^{(n)})=0\quad x\in I. Given two solutions u:J⊂R→Ru:J\subset \mathbb {R} \to \mathbb {R} and v:I⊂R→Rv:I\subset \mathbb {R} \to \mathbb {R} , uu is called an extension of vv if I⊂JI\subset J and u(x)=v(x)x∈I. 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 R\mathbb {R} is called a global solution. A general solution of an nn th-order equation is a solution containing nn 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′=−sgn(y)∣y∣,  y(0)=1y'=-{\text{sgn}}(y){\sqrt {|y|}},\,\,y(0)=1 Admits the finite duration solution: y(x)=14(1−x2+∣1−x2∣)2y(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.