Preface

It is important to know when some equations have solutions with similar properties. This section gives classification of linear and nonlinear equations. It also provides methods how to convert linear differential equations to canonical forms.
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Glossary

Canonical forms

The consideration of initial value problems for second-order ordinary differential equations is motivated by a number of physical problems in various fields. Therefore, we start with reducing linear various coefficient differential equations to canonical equations.
A linear differential equation of second order that does not contain the first derivative,
\[ \frac{{\text d}^2 y}{{\text d} x^2} + q(x)\, y(x) = f(x) , \]
is said to be in the normal form.

 

Linear differential equations

A linear homogeneous differential equation of the second order
\begin{equation} y'' + a(x)\,y' + b(x)\, y = 0 \label{Eq4can.1} \end{equation}
can be reduced to a normal form with either change of independent variable or dependent variable.

Transformation 1: if the independent variable is changed as

\begin{equation} t = \int_{x_0}^x \exp \left\{ -\int_{x_0}^r a(z)\,{\text d}z \right\} {\text d}r , \label{eqn:ode1:tr1} \end{equation}
then Eq.\eqref{Eq4can.1} becomes
\begin{equation} \frac{{\text d}^2 y}{{\text d} t^2} + b(x(t))\,\exp \left\{ -\int_{x_0}^x a(x)\,{\text d}z \right\} y = 0. \label{trans:2:1:c} \end{equation}
Example: For the ordinary differential equation \begin{equation*} y''-\frac{3x}{1-x^2}y'+\frac{7}{1-x^2}y=0, \end{equation*} the change of variables in \eqref{eqn:ode1:tr1} becomes \( t=x /\sqrt{1-x^2} \) and the equation corresponding to \eqref{trans:2:1:c} is \begin{equation*} \frac{{\text d}^2 y}{{\text d}t^2} + \frac{7}{(1+t^2)^2}\, y=0. \end{equation*}    ■

Transformation 2: if in Eq.\eqref{Eq4can.1} the expression \begin{equation} \frac{b'+2ab}{b^{3/2} } \label{trans:2:1:d} \end{equation} is found to be a constant, then the change of independent variable given by \begin{equation} z=C\int \sqrt{b(x)}\,{\text d} x, \label{trans:2:1:b} \end{equation} where C is an arbitrary constant, will reduce equation \eqref{Eq4can.1} to an equation with constant coefficients. Moreover, if the expression in equation \eqref{trans:2:1:d} is not constant, then no change of independent variable alone will reduce equation \eqref{Eq4can.1} to an equation with constant coefficients.

Example: Given the equation \begin{equation*} xy''+(8x^2-1)y'+20x^3y=0, \label{trans:2:1:f} \end{equation*} we note that 𝑎(x) =8x -1/x and b(x) =20 x². Hence, the expression in equation \eqref{trans:2:1:d} becomes \begin{equation*} \frac{b'+2ab}{b^{3/2} } =\frac{40x+40x^2(8x-x^{-1})}{20^{3/2}x^3} =\frac{320x^3}{20^{3/2}x^3} =\frac{320}{20^{3/2}} = \textrm{constant}. \end{equation*} Therefore, if the independent variable is changed by \( \displaystyle z=C\int \sqrt{20} x\,{\text d} x, \) then the previous equation, written in terms of z, will be a constant coefficient differential equation. A natural choice for C is \( C=2/\sqrt{20} \) so that the transformation becomes z = x². Using this new variable in equation above results in the equation \begin{equation*} \frac{{\text d}^2 y}{{\text d} z^2} + 4\,\frac{{\text d}y}{{\text d}z} +5\,y = 0, \end{equation*} which has the solution \( \displaystyle y=e^{-2z} \left( A\cos z+B\sin z \right) , \) where A and B are arbitrary constants. Hence, the general solution to the given equation is \begin{equation*} y= \left( A\cos x^2+B\sin x^2 \right) \exp \! \left\{ -2x^2 \right\} . \end{equation*}    ■

Transformation 3: if the dependent variable is changed by \begin{equation} y(x)=u(x) \exp \! \left\{ -\frac{1}{2}\int^x a(z)\,{\text d} z \right\} , \end{equation} then equation \eqref{Eq4can.1} becomes \begin{equation} u''+q(x) u=0, \label{trans:2:1:m} \end{equation} where \begin{equation} q(x)= b-\frac14 a^2-\frac{1}{2}\,\frac{{\text d}a}{{\text d}x} . \label{trans:2:1:n} \end{equation}

Example: Suppose we wish to solve the equation \begin{equation} \frac{{\text d}^2 y}{{\text d} x^2} - \frac{2}{x}\,\frac{{\text d}y}{{\text d}x} + \left( a^2+\frac{2}{x^2} \right) y =0, \label{trans:2:1:q} \end{equation} in which 𝑎 is a constant. We find that using equation \eqref{trans:2:1:n}) \begin{equation} q(x) = \surround{ a^2+\frac{2}{x^2} } -\frac14\frac{4}{x^2} -\frac{1}{2} \frac{2}{x^2} =a^2. \end{equation} Now, we know the solution of \begin{equation*} \frac{{\text d}^2 y}{{\text d}t^2} + a^2 v=0 \label{trans:2:1:s} \end{equation*} to be \( \displaystyle v(x)=A\cos ax+B\sin ax , \) where A and B are arbitrary constants.    ■

 

Exact differential equations

A differential equation
\begin{equation} a_2 (x,y,y' )\,y'' + a_1 (x,y,y')\,y' + a_0 (x,y,y')\, y = f(x) \label{Eq4exact.1} \end{equation}
is called the exact second order differential equation if there exist a potential function Ψ(x,y,y') such that Eq.\eqref{Eq4exact.1} can be written as full differential:
\begin{equation*} \frac{\text d}{{\text d}x}\,\Psi (x,y,y') = f(x) \qquad \Longleftrightarrow \qquad \frac{\partial \Psi (x,y,y')}{\partial y'}\, y'' + \frac{\partial \Psi (x,y,y')}{\partial y}\, y' + \frac{\partial \Psi (x,y,y')}{\partial x}\, y = f(x). \end{equation*}
This is equivalent that
\begin{equation} a_0 (x,y,y') = \frac{\partial \Psi (x,y,y')}{\partial x} , \qquad a_1 (x,y,y') = \frac{\partial \Psi (x,y,y')}{\partial y} , \qquad a_2 (x,y,y') = \frac{\partial \Psi (x,y,y')}{\partial y'} . \label{Eq4exact.2} \end{equation}
Theorem: Let the functions \( a_0 (x,y,y'), \ a_1 (x,y,y'), \ a_2 (x,y,y'), \ \frac{\partial a_0}{\partial y'}, \ \frac{\partial a_0}{\partial y}, \ \frac{\partial a_1}{\partial y'}, \ \frac{\partial a_1}{\partial x}, \ \frac{\partial a_2}{\partial x}, \ \frac{\partial a_1}{\partial y} \) be continuous functions in a simply connected region R ⊆ ℝ³. Then Eq.\eqref{Eq4exact.1} is exact if and only if
\begin{equation} \frac{\partial a_2}{\partial y} = \frac{\partial a_1}{\partial y'} , \quad \frac{\partial a_2}{\partial x} = \frac{\partial a_0}{\partial y'} , \quad \frac{\partial a_1}{\partial x} = \frac{\partial a_0}{\partial y} . \label{Eq4exact.3} \end{equation}
In this case, the potential function can be determined by integration:
\begin{equation} \Psi (x,y,y') = \int_{x_0}^x a_0 \left( \alpha , y, y' \right) {\text d} \alpha + \int_{y_0}^y a_1 \left( x_0 , \beta , y' \right) {\text d} \beta + \int_{y'_0}^{y'} a_2 \left( x_0 , y_0 , \gamma \right) {\text d} \gamma . \label{Eq4exact.4} \end{equation}
Example: The second order nonlinear differential equation
\[ 3\epsilon\,y'' + y\,y' = 0 \]
is exact. We find its potential function:
\[ \Psi (x,y,y' ) = \int_0^y \beta\,{\text d} \beta + 3\epsilon \int_0^{y'} {\text d} \gamma = \frac{y^2}{2} + 3\epsilon\, y' . \]
Upon integration, we reduce the second order differential equation to the first order equation:
\[ \Psi (x,y,y' ) = \frac{y^2}{2} + 3\epsilon\, y' = c. \]
   ■
Example: Consider the initial value problem
\[ \begin{cases} y'' + 12x\,y^3 y' + \left( 3y^4 -1 \right) y = 0, \\ y(0) = 3, \quad y' (0) = 0. \end{cases} \]
Using formula \eqref{Eq4exact.4}, we obtain the potential function:
\begin{align*} \Psi (x,y,y' ) &= \int_0^x \left( 3y^4 -1 \right) {\text d} \alpha + \int_3^y \left( 12x\,y^3 \right)_{x=0} {\text d} \beta + \int_0^{y'} {\text d} \gamma \\ &= \left( 3y^4 -1 \right) x + y' . \end{align*}
This allows us to reduce the given equation to the differential equation of the first order:
\[ y' + \left( 3y^4 -1 \right) x = 0 , \qquad y(0) = 3. \]
   ■
Consider the linear differential equation of the second order
\begin{equation} \frac{\text d}{{\text d}x} \left[ p(x) \,\frac{{\text d}y}{{\text d}x} \right] + r(x)\,\frac{{\text d}y}{{\text d}x} + q(x)\,y = f(x) , \qquad x > x_0 \label{Eq4exact.5} \end{equation}
where p ∈ C¹, and r, q, f are continuous functions. Multiplying both sides of Eq.\eqref{Eq4exact.5} by \( \displaystyle \xi_1 (x) = \exp \left\{ \int \frac{q(x)}{r(x)}\,{\text d} x \right\} , \) we get
\[ \xi_1 (x) \,\frac{\text d}{{\text d}x} \left[ p(x) \,\frac{{\text d}y}{{\text d}x} \right] + \xi_1 (x) \,r(x)\,\frac{{\text d}y}{{\text d}x} + \xi_1 (x) \,q(x)\,y = \xi_1 (x) \,f(x) . \]
Taking into account \( \displaystyle \xi_1' (x)\,r(x) = \xi_1 (x) \,q(x) , \) we obtain
\[ \xi_1 (x) \,\frac{\text d}{{\text d}x} \left[ p(x) \,\frac{{\text d}y}{{\text d}x} \right] + r(x)\,\frac{{\text d}}{{\text d}x} \left( \xi_1 (x) \,y \right) = \xi_1 (x) \,f(x) . \]
Then
\begin{equation*} \frac{\text d}{{\text d}x} \left[ p(x) \,\frac{{\text d}y}{{\text d}x} \right] + \frac{r(x)}{\xi_1 (x)} \,\frac{{\text d}}{{\text d}x} \left( \xi_1 (x) \,y \right) = f(x) . \end{equation*}
If we set z = ξ1y, we get
\begin{equation} \frac{\text d}{{\text d}x} \left[ \frac{p(x)}{\xi_1 (x)} \,\frac{{\text d}z}{{\text d}x} \right] + \frac{{\text d}}{{\text d}x} \left( p(x) \left( \frac{1}{\xi_1 (x)} \right)' z \right) + \frac{r(x)}{\xi_1 (x)} \,\frac{{\text d}z}{{\text d}x} = f(x) . \label{Eq4exact.6} \end{equation}
If we choose
\[ p(x) \left( \frac{1}{\xi_1 (x)} \right)' = c \qquad \Longleftrightarrow \qquad c\,r(x) + p(x) \,\exp\left\{ \int \frac{q(x)}{r(x)} \, {\text d} x \right\} = 0 \]
then Eq.\eqref{Eq4exact.6} becomes
\begin{equation} \frac{\text d}{{\text d}x} \left[ s(x) \,\frac{{\text d}z}{{\text d}x} \right] + t(x)\,\frac{{\text d}z}{{\text d}x} = f(x) , \qquad s(x) = \frac{p(x)}{\xi_1 (x)} , \quad t(x) = \frac{r(x)}{\xi_1 (x)} + c. \label{Eq4exact.7} \end{equation}
Multiplying both sides by \( \displaystyle \xi_2 (x) = \exp \left\{ \int \frac{t(x)}{s(x)}\,{\text d} x \right\} , \) we get the exact equation
\[ \frac{\text d}{{\text d}x} \left[ \xi_1 (x) s(x)\,\frac{{\text d}z}{{\text d}x} \right] = \xi_2 (x)\, f(x) . \]
Example: Consider the confluent hypergeometric equation that was introduced by the German mathematician Ernst Kummer in 1837:
\[ \frac{{\text d}^2 y}{{\text d} x^2} + \frac{b-x}{x} \,\frac{{\text d}y}{{\text d}x} - \frac{a}{x}\, y = 0, \qquad y(0) =1, \quad y' (0) = -\frac{1}{b} . \]
For 𝑎 = 1, we have
\[ p(x) = 1 , \quad r(x) = \frac{b-x}{x}, q(x) = -\frac{a}{x}, \quad \xi_1 (x) = \frac{1}{b-x} , \quad \xi_2 (x) = \frac{x^b}{b-x}\, e^{-x}, \quad s(x) = b-x, \quad t(x) = 1 - \frac{(b-x)^2}{x} . \]
Then the exact solution becomes \( \displaystyle y(x) = \frac{b-x}{b} . \)    ■
Example: Consider the initial value problem for the Legendre equation
\[ \left( 1- x^2 \right) y'' -2x\,y' + 2\,y = 2, \qquad y(-1) = 2, \quad y' (-1) = -1. \]
Be aware that the second initial condition y(-1) = -1 is redundant and can be dropped because it is satisfied automatically. Recall that the Legendre equation has two regular singular points x = ±1 and two initial conditions are needed only when the initial point is regular. Using
\[ \xi_1 (x) = x^{-1}, \quad \xi_2 (x) = x\left( 1- x^2 \right) , \quad s(x) = x, \quad t(x) = 1 - \frac{2x}{1- x^2} , \]
we obtain the exact solution y = 1 - x.    ■

 

Self-adjoint differential equations

If both sides of the equation \begin{equation} y''+a(x)y'+b(x)y=c(x) \label{trans:2:1:w} \end{equation} are multiplied by \begin{equation*} p(x)=\exp \! \left\{ \int_{x_0}^x a(z)\,{\text d} z \right\} , \end{equation*} then equation \eqref{trans:2:1:w} is put in the formally self-adjoint form \begin{equation} \label{EqCanonic.19} \frac{\text d}{{\text d}x} \left( p(x)\, \frac{{\text d}y}{{\text d}x} \right) +q(x)\,y=r(x), \end{equation} where \begin{equation} \begin{split} q(x) &= p(x) b(x), \\ r(x) &= p(x) c(x). \\ \end{split} % \label{eqn:...} \end{equation} If the independent variable in equation \eqref{EqCanonic.19} is changed from x to s by \( \displaystyle s= \int\frac{{\text d} x}{p(x)} , \) and if p(x) > 0 for x >x0, and \( \displaystyle \int_{x_0}^{\infty}\frac{\textd x}{p(x)} =\infty , \) then equation \eqref{EqCanonic.19} becomes \begin{equation} \label{EqCanonic.21} \frac{{\text d}^2 y}{{\text d}s^2} + p(s)\,q(s)\, y = 0. \end{equation} Note that, as x → ∞, the variable s → ∞
Example: For the ordinary differential equation \( \displaystyle (x\,y')'+y=0, \) we identify p(x) = x, q(x) = 1, and x0 = 0. Hence, the change of variable s = ln x results in \( \displaystyle y_{ss}+e^s y=0. \)    ■

If the dependent variable in equation \eqref{EqCanonic.19} is changed from y(x) to u(x) by \begin{equation} u(x)=\sqrt{p(x)}\,y(x), \end{equation} then equation \eqref{EqCanonic.19} becomes \begin{equation} \frac{{\text d}^2 u}{{\text d}x^2} + \left[ \frac{q}{p} - \frac{1}{2}\,\frac{\text d}{{\text d}x} \left( \frac{p'}{p} \right) - \frac{1}{4} \left( \frac{p'}{p} \right)^2 \right] u = 0. \end{equation}

Example: For the ordinary differential equation \( \displaystyle (x\,y')'+y=0, \) we identify p(x) = x, q(x) = 1, and x0 = 0. Hence, the change of variable \( \displaystyle u(x)=\sqrt{x}\,y(x) \) results in \( \displaystyle u'' + \frac{4x+1}{4x^2} \,u=0. \)    ■

If the range of interest for equation \eqref{EqCanonic.19} is x0 < x < ∞ and if the independent and dependent variables are changed by \begin{equation} \begin{split} t = & \int_{x_0}^x \sqrt{ \frac{|q(z)|}{p(z)} }\,{\text d} z, \\ u(t)= & \left[ p(x) \, |q(x)| \right]^{1/4} y(x), \end{split} \end{equation} then equation \eqref{EqCanonic.19} becomes \begin{equation} \label{EqCanonic.25} \frac{{\text d}^2 u}{{\text d} t^2} +\left[ \pm 1+R(t) \right] u(t)=0, \end{equation} where \begin{equation} R(t) = \left. p^{1/4}\,|q|^{-3/4}\,\frac{{\text d}}{{\text d}x} \left[ p(x)\,|q(x)|\right]^{-1/4} \right\vert_{x=x(t)}, \end{equation} and the plus (minus) sign is taken in equation \eqref{EqCanonic.25} if q(x) > 0 (q(x) < 0).

>

This transformation is sometimes called the Liouville--Green transformation, although it is virtually identical to the Liouville transformation.

If the independent and dependent variables are changed in equation \eqref{EqCanonic.19} by \begin{equation} \begin{split} y(x)= & a(x) w(t), \\ t = & \int^x b(z) \,{\text d} z, \end{split} \end{equation} then equation \eqref{EqCanonic.19} becomes \begin{equation} \frac{b}{a} \,\frac{\text d}{{\text d}t}\, \left( p a^2 b\,\frac{{\text d}w}{{\text d}t} \right) +L[a]\,w=0. \label{trans:solode2:k} \end{equation} Note that the operator L[ · ] is defined by equation \eqref{EqCanonic.19}. If b(z) is chosen to be \begin{equation} b(z)=\frac{1}{p(z)a^2(z)}, \end{equation} then equation \eqref{trans:solode2:k} simplifies to \( \displaystyle \frac{1}{p a^3} \,\frac{{\text d}^2 w}{{\text d}t^2} +L[a]\,w=0. \)

 

Painlevé classification

The classification of second order nonlinear differential equations,
\[ y'' = R\left( y' , y , x \right) , \]
(with R a rational function) was started by the French mathematician and statesman Paul Painlevé (1863--1933) and his students at the turn of twentieth century. Painlevé found fifty nonlinear differential equations in a polynomial form that have general solutions without critical movable points. These solutions are called Painlevé transcendents that are characterized by so called the Painlevé property (the only movable singularities are poles), but which are not generally solvable in terms of elementary functions. However the Painlevé property is just the necessary condition for integrability of differential equations and there is a problem of finding the general solution of nonlinear differential equations.

Out of fifty canonical Painlevé equations, forty-four of the them are reducible to just six equations requiring the introduction of new special functions to solve them. Explicit transformations and relevant exact solutions admitted by the Painlevé equations first appeared in the Soviet mathematical school from Minsk (its summary can be found in reviews by N.P.Erugin and V.I. Gromak). These six equations, traditionally called Painlevé I--VI, are as follows:

  1. (P. Painlevé): \( \displaystyle \frac{{\text d}^2 y}{{\text d}x^2} = 6\, y^2 + x . \)
  2. (P. Painlevé): \( \displaystyle \frac{{\text d}^2 y}{{\text d}x^2} = 2\, y^3 + x\,y + \alpha . \)
  3. (P. Painlevé): \( \displaystyle \frac{{\text d}^2 y}{{\text d}x^2} = x \left( \frac{{\text d}y}{{\text d}x} \right)^2 + -y\,\frac{{\text d}y}{{\text d}x} + \delta\,x + \beta \,y + \alpha \,y^3 + \gamma x\,y^4 . \)
  4. (B. Gamblier): \( \displaystyle \frac{{\text d}^2 y}{{\text d}x^2} = \frac{1}{2} \left( \frac{{\text d}y}{{\text d}x} \right)^2 + \beta + 2 \left( x^2 - \alpha \right) y^2 + 4x\,y^3 + \frac{3}{2}\, y^4 . \)
  5. (B. Gamblier): \( \displaystyle y\,\frac{{\text d}^2 y}{{\text d}x^2} = \left( \frac{1}{2y} + \frac{1}{y-1} \right) \left( \frac{{\text d}y}{{\text d}x} \right)^2 - \frac{1}{x}\, \frac{{\text d}y}{{\text d}x} + \frac{(y-1)^2}{x^2} \left( \alpha\,y + \frac{\beta}{y} \right) + \gamma\, \frac{y}{x} + \delta \, \frac{y\,(y+1)}{y-1} . \)
  6. (R. Fuchs): \( \displaystyle \frac{{\text d}^2 y}{{\text d}x^2} = \frac{1}{2} \left( \frac{1}{y} + \frac{1}{y-1} + \frac{1}{y-x} \right) \left( \frac{{\text d}y}{{\text d}x} \right)^2 - \left( \frac{1}{x} + \frac{1}{x-1} + \frac{1}{y-x} \right) \frac{{\text d}y}{{\text d}x} + \frac{y\,(y-1)\,(y-x)}{x^2 \left( x-1 \right)^2} \left[ \alpha + \beta \,\frac{x}{y^2} + \gamma \,\frac{x-1}{(y-1)^2} + \delta\,\frac{x\,(x-1)}{(y-x)^2} \right] . \)

For certain choices of the parameters, PII--V admit one-parameter families of solutions expressible in terms of the classical transcendental functions: Airy, Bessel, Weber--Hermite, and Whittaker. There is a deep connection between ordinary differential equations of Painlevé type and the partial differential equations (see M.J. Ablowitz).

 

  1. Ablowitz, M.J., Ramani, A., and Segur, H., Lettere al Nuovo Cimento, 1978, Vol. 23, pp. 333; Journal of Mathematical Physics, 1980, Vol. 21, pp. 715; 1980, Vol. 21, pp. 1006.
  2. Erugin, N.P., Theory of moving singular points of second order equations. I, Differential Equations, 1976, Vol. 12, Issue 3, pp. 387--416.
  3. Erugin, N.P., Theory of moving singular points of second order equations. II, Differential Equations, 1976, Vol. 12, Issue 4, pp. 579--598.
  4. Fokas, A.S. and Ablowitz, M.J., On a unified approach to transformations and elementary solutions of Painlevé equations, Journal of Mathematical Physics, 1982, Vol. 23, June, pp. 2033. https://doi.org/10.1063/1.525260
  5. Gambier, B., Surles équations différentielles du second ordre et du premier degré dont l’intégrale générale est à points critique fixés, Acta mathematica, Vol. 33 (1910) 1–55.
  6. Garnier, R., Sur des équations différentielles du troisiéme ordre dont l’intégrale est uniform et sur une classe d’équations nouvelles d’ordre supérieur dont l’intégrale générale a ses point critiques fixés, Annales Scientifiques de l'École Normale Supérieure, Sér. 3 29 (1912) 1–126.
  7. Gromak, V.I., Single-parameter families of solutions of Painlevé equations, Differential Equations, 1978, Vol. 14, Issue 12, pp. 2131--2135.
  8. Hu, H., Guo, Y.-J., Xu, D.-Q., Application of the Liouville–Green transformation to freevibrations of non-homogeneous rectangular membranes, Arch Appl Mech (2014) 84: pp. 999–1006; doi: 10.1007/s00419-014-0844-z
  9. Kudryashov, N.A., Sinelshchikov, D.I., Nonlinear differential equations of the second, third and fourth order with exact solutions, Applied Mathematics and Computation, 2012, Vol. 218, pp. 10454--10467.
  10. Painlevé, P., Mémoire sur les équations différentielles dont l’intégrale générale est uniforme, Bulletin de la Société Mathématique de France, Vol. 28 (1900) 201–261.
  11. P. Painlevé, Surles équations différentielles du second ordre et d’ordre supérieur dont l’intégrale gènèrale est uniforme, Acta mathematica, Vol. 25 (1902) 1–85.
  12. P. Painlevé, Sur les equations differentielles du second order a points critiques fixes, Comptes Rendus Mathématique Acad. Sci. Paris 143 (1906) 1111–1117.
  13. Zwillinger, D. and Dobrushkin, V., Handbook on Differential Equations, 2020, CRC Press.

 

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