Special solution

A special solution of an ordinary differential equation is a solution in any neighborhood of each point of which the uniqueness of the solution of the Cauchy problem for this equation is violated.

Content

Definition

Consider the equation

F(x,y,y')=0,\ \ \ (1)

{\ displaystyle F (x, y, y ') = 0, \ \ \ (1)}

Where $F(x,y,p)$ ${\ displaystyle F (x, y, p)}$ - $C^{1}$ ${\ displaystyle C ^ {1}}$ -Smooth function in some area $G\subseteq {\mathbb {R} ^{3}}_{(x,y,p)}$ ${\ displaystyle G \ subseteq {\ mathbb {R} ^ {3}} _ {(x, y, p)}}$ . Solution of equation (1) $y=\psi (x),x\in \mathrm {I} \subseteq \mathbb {R} ,$ ${\ displaystyle y = \ psi (x), x \ in \ mathrm {I} \ subseteq \ mathbb {R},}$ called a special solution if every point $(x_{0},\psi (x_{0})),x_{0}\in \mathrm {I} ,$ ${\ displaystyle (x_ {0}, \ psi (x_ {0})), x_ {0} \ in \ mathrm {I},}$ the corresponding integral curve is the point of local non-uniqueness of the solution of the Cauchy problem with the initial condition $y(x_{0})=\psi (x_{0})$ ${\ displaystyle y (x_ {0}) = \ psi (x_ {0})}$ .

Properties

Special solution $y=\psi (x),x\in \mathrm {I}$ ${\ displaystyle y = \ psi (x), x \ in \ mathrm {I}}$ , equations (1) geometrically means that the integral curve $y=\psi (x)$ ${\ displaystyle y = \ psi (x)}$ at each point it touches some other integral curve of equation (1) and does not coincide with it in some neighborhood of this point, that is, it is the envelope of the family of integral curves of equation (1).

A special solution of equation (1), if it exists, is always the discriminant curve of this equation ^[1] (more precisely, it is always part of the discriminant curve: the latter can have several branches with different properties). The converse is not true: the discriminant curve is not necessarily a solution to the equation (and if it is, it is not necessarily special) ^[2] . So, for example, the discriminant curve of the Cibrario equation $(y')^{2}=x$ ${\ displaystyle (y ') ^ {2} = x}$ is not a solution, but a geometric locus of the points of return of its integral curves.

Therefore, to find practical solutions to the equation, you must first find its discriminant curve, and then check whether it (each of its branches, if there are several) is a special solution to equation (1) or not ^[2] .

Examples

Special solutions of differential equations (thick lines): Claireau equations (left) and equations

(y')^{2}=y

{\ displaystyle (y ') ^ {2} = y}

(on right).

1. Simple examples of differential equations having special solutions are the Claireau equation and the equation $(y')^{2}=y$ ${\ displaystyle (y ') ^ {2} = y}$ whose nonsingular solutions are given by the formula $y={\frac {1}{4}}(x-c)^{2}$ ${\ displaystyle y = {\ frac {1} {4}} (xc) ^ {2}}$ with constant integration $c$ ${\ displaystyle c}$ , and the special solution has the form $y=0$ ${\ displaystyle y = 0}$ .

2. The discriminant curve of the equation $(y')^{2}=4y^{3}(1-y)$ ${\ displaystyle (y ') ^ {2} = 4y ^ {3} (1-y)}$ consists of two disjoint branches: $y=1$ ${\ displaystyle y = 1}$ and $y=0$ ${\ displaystyle y = 0}$ . Both are solutions to this equation. However, the first of them is a special solution, and the second is not: at each point of the line $y=1$ ${\ displaystyle y = 1}$ it touches some other integral curve of this equation, and to the line $y=0$ ${\ displaystyle y = 0}$ integral curves only approach asymptotically for $x\to \infty$ ${\ displaystyle x \ to \ infty}$ ^[3] .

Notes

↑ The discriminant curve of equation (1) is a curve on the plane of variables $(x,y)$ ${\ displaystyle (x, y)}$ defined by equations $F(x,y,p)=F_{p}(x,y,p)=0$ ${\ displaystyle F (x, y, p) = F_ {p} (x, y, p) = 0}$ .
↑ ¹ ² Filippov A.F. Introduction to the theory of differential equations. - M.: URSS, 2007, ch. 2, paragraph 8.
↑ Filippov A.F. Introduction to the theory of differential equations. - M.: URSS, 2007, ch. 2, paragraph 8, example 5.

Literature

Arnold V.I. Additional chapters of the theory of ordinary differential equations. - M.: Science, 1978.
Arnold V.I. Geometric methods in the theory of ordinary differential equations. - Izhevsk: Publishing house of the Udmurt state. University, 2000.
Romanko V.K. Course of differential equations and calculus of variations. - M .: Fizmatlit, 2001.
Filippov A.F. Introduction to the theory of differential equations. - M.: URSS, 2004, 2007.

[1] The discriminant curve of equation (1) is a curve on the plane of variables $(x,y)$ ${\ displaystyle (x, y)}$ defined by equations $F(x,y,p)=F_{p}(x,y,p)=0$ ${\ displaystyle F (x, y, p) = F_ {p} (x, y, p) = 0}$ .

[autogenerated1-2] ¹ ² Filippov A.F. Introduction to the theory of differential equations. - M.: URSS, 2007, ch. 2, paragraph 8.

[3] Filippov A.F. Introduction to the theory of differential equations. - M.: URSS, 2007, ch. 2, paragraph 8, example 5.