2.32.22 Problem 204

Internal problem ID [13864]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-6
Problem number : 204
Date solved : Friday, December 19, 2025 at 05:14:31 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x \left (x -1\right ) \left (-a +x \right ) y^{\prime \prime }+\left (\left (\alpha +\beta +1\right ) x^{2}-\left (\alpha +\beta +1+a \left (\gamma +d \right )-a \right ) x +a \gamma \right ) y^{\prime }+\left (\alpha \beta x -q \right ) y&=0 \\ \end{align*}

2.32.22.1 ✓ Maple. Time used: 0.118 (sec). Leaf size: 82

ode:=x*(x-1)*(x-a)*diff(diff(y(x),x),x)+((alpha+beta+1)*x^2-(alpha+beta+1+a*(gamma+d)-a)*x+a*gamma)*diff(y(x),x)+(alpha*beta*x-q)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \operatorname {HeunG}\left (a , q , \alpha , \beta , \gamma , \frac {a \left (d -1\right )}{a -1}, x\right )+c_2 \,x^{1-\gamma } \operatorname {HeunG}\left (a , q -\left (-1+\gamma \right ) \left (a \left (d -1\right )+\alpha +\beta -\gamma +1\right ), \beta +1-\gamma , \alpha +1-\gamma , -\gamma +2, \frac {a \left (d -1\right )}{a -1}, x\right ) \]

Maple trace

Methods for second order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
checking if the LODE has constant coefficients 
checking if the LODE is of Euler type 
trying a symmetry of the form [xi=0, eta=F(x)] 
checking if the LODE is missing y 
-> Trying a Liouvillian solution using Kovacics algorithm 
<- No Liouvillian solutions exists 
-> Trying a solution in terms of special functions: 
   -> Bessel 
   -> elliptic 
   -> Legendre 
   -> Kummer 
      -> hyper3: Equivalence to 1F1 under a power @ Moebius 
   -> hypergeometric 
      -> heuristic approach 
      -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius 
   -> Mathieu 
      -> Equivalence to the rational form of Mathieu ODE under a power @ Moebi\ 
us 
trying a solution in terms of MeijerG functions 
-> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power \ 
@ Moebius 
<- Heun successful: received ODE is equivalent to the  HeunG  ODE, case  a <> 0\ 
, e <> 0, g <> 0, c = 0
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x \left (x -1\right ) \left (x -a \right ) \left (\frac {d^{2}}{d x^{2}}y \left (x \right )\right )+\left (\left (\alpha +\beta +1\right ) x^{2}-\left (\alpha +\beta +1+a \left (\gamma +d \right )-a \right ) x +a \gamma \right ) \left (\frac {d}{d x}y \left (x \right )\right )+\left (\alpha \beta x -q \right ) y \left (x \right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & \frac {d^{2}}{d x^{2}}y \left (x \right ) \\ \bullet & {} & \textrm {Isolate 2nd derivative}\hspace {3pt} \\ {} & {} & \frac {d^{2}}{d x^{2}}y \left (x \right )=\frac {\left (\alpha \beta x -q \right ) y \left (x \right )}{x \left (x -1\right ) \left (-x +a \right )}-\frac {\left (a d x +a \gamma x -x^{2} \alpha -x^{2} \beta -a \gamma -a x +\alpha x +\beta x -x^{2}+x \right ) \left (\frac {d}{d x}y \left (x \right )\right )}{x \left (x -1\right ) \left (-x +a \right )} \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y \left (x \right )\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE; ODE is linear}\hspace {3pt} \\ {} & {} & \frac {d^{2}}{d x^{2}}y \left (x \right )+\frac {\left (a d x +a \gamma x -x^{2} \alpha -x^{2} \beta -a \gamma -a x +\alpha x +\beta x -x^{2}+x \right ) \left (\frac {d}{d x}y \left (x \right )\right )}{x \left (x -1\right ) \left (-x +a \right )}-\frac {\left (\alpha \beta x -q \right ) y \left (x \right )}{x \left (x -1\right ) \left (-x +a \right )}=0 \\ \square & {} & \textrm {Check to see if}\hspace {3pt} x_{0}\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & \circ & \textrm {Define functions}\hspace {3pt} \\ {} & {} & \left [P_{2}\left (x \right )=\frac {a d x +a \gamma x -x^{2} \alpha -x^{2} \beta -a \gamma -a x +\alpha x +\beta x -x^{2}+x}{x \left (x -1\right ) \left (-x +a \right )}, P_{3}\left (x \right )=-\frac {\alpha \beta x -q}{x \left (x -1\right ) \left (-x +a \right )}\right ] \\ {} & \circ & x \cdot P_{2}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =0 \\ {} & {} & \left (x \cdot P_{2}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}0}}}=\gamma \\ {} & \circ & x^{2}\cdot P_{3}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =0 \\ {} & {} & \left (x^{2}\cdot P_{3}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}0}}}=0 \\ {} & \circ & x =0\textrm {is a regular singular point}\hspace {3pt} \\ & {} & \textrm {Check to see if}\hspace {3pt} x_{0}\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & {} & x_{0}=0 \\ \bullet & {} & \textrm {Multiply by denominators}\hspace {3pt} \\ {} & {} & x \left (x -1\right ) \left (-x +a \right ) \left (\frac {d^{2}}{d x^{2}}y \left (x \right )\right )+\left (a d x +a \gamma x -x^{2} \alpha -x^{2} \beta -a \gamma -a x +\alpha x +\beta x -x^{2}+x \right ) \left (\frac {d}{d x}y \left (x \right )\right )+\left (-\alpha \beta x +q \right ) y \left (x \right )=0 \\ \bullet & {} & \textrm {Assume series solution for}\hspace {3pt} y \left (x \right ) \\ {} & {} & y \left (x \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +r} \\ \square & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{m}\cdot y \left (x \right )\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =0..1 \\ {} & {} & x^{m}\cdot y \left (x \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +r +m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k -m \\ {} & {} & x^{m}\cdot y \left (x \right )=\moverset {\infty }{\munderset {k =m}{\sum }}a_{k -m} x^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{m}\cdot \left (\frac {d}{d x}y \left (x \right )\right )\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =0..2 \\ {} & {} & x^{m}\cdot \left (\frac {d}{d x}y \left (x \right )\right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) x^{k +r -1+m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1-m \\ {} & {} & x^{m}\cdot \left (\frac {d}{d x}y \left (x \right )\right )=\moverset {\infty }{\munderset {k =-1+m}{\sum }}a_{k +1-m} \left (k +1-m +r \right ) x^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{m}\cdot \left (\frac {d^{2}}{d x^{2}}y \left (x \right )\right )\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =1..3 \\ {} & {} & x^{m}\cdot \left (\frac {d^{2}}{d x^{2}}y \left (x \right )\right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) \left (k +r -1\right ) x^{k +r -2+m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +2-m \\ {} & {} & x^{m}\cdot \left (\frac {d^{2}}{d x^{2}}y \left (x \right )\right )=\moverset {\infty }{\munderset {k =-2+m}{\sum }}a_{k +2-m} \left (k +2-m +r \right ) \left (k +1-m +r \right ) x^{k +r} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & -a a_{0} r \left (\gamma -1+r \right ) x^{-1+r}+\left (-a a_{1} \left (1+r \right ) \left (\gamma +r \right )+a_{0} \left (a d r +a \gamma r +a \,r^{2}-2 a r +\alpha r +\beta r +r^{2}+q \right )\right ) x^{r}+\left (\moverset {\infty }{\munderset {k =1}{\sum }}\left (-a a_{k +1} \left (k +1+r \right ) \left (\gamma +k +r \right )+a_{k} \left (a d k +a d r +a \gamma k +a \gamma r +a \,k^{2}+2 a k r +a \,r^{2}-2 a k -2 a r +\alpha k +\alpha r +\beta k +\beta r +k^{2}+2 k r +r^{2}+q \right )-a_{k -1} \left (\beta +k -1+r \right ) \left (\alpha +k -1+r \right )\right ) x^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & -a r \left (\gamma -1+r \right )=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{0, 1-\gamma \right \} \\ \bullet & {} & \textrm {Each term must be 0}\hspace {3pt} \\ {} & {} & -a a_{1} \left (1+r \right ) \left (\gamma +r \right )+a_{0} \left (a d r +a \gamma r +a \,r^{2}-2 a r +\alpha r +\beta r +r^{2}+q \right )=0 \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & -a a_{k +1} \left (k +1+r \right ) \left (\gamma +k +r \right )+a_{k} \left (\left (k +r \right ) \left (k +d +r +\gamma -2\right ) a +k^{2}+\left (2 r +\beta +\alpha \right ) k +r^{2}+\left (\alpha +\beta \right ) r +q \right )-a_{k -1} \left (\beta +k -1+r \right ) \left (\alpha +k -1+r \right )=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1 \\ {} & {} & -a a_{k +2} \left (k +2+r \right ) \left (\gamma +k +1+r \right )+a_{k +1} \left (\left (k +1+r \right ) \left (k -1+d +r +\gamma \right ) a +\left (k +1\right )^{2}+\left (2 r +\beta +\alpha \right ) \left (k +1\right )+r^{2}+\left (\alpha +\beta \right ) r +q \right )-a_{k} \left (\beta +k +r \right ) \left (\alpha +k +r \right )=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +2}=\frac {a d k a_{k +1}+a d r a_{k +1}+a \gamma k a_{k +1}+a \gamma r a_{k +1}+a \,k^{2} a_{k +1}+2 a k r a_{k +1}+a \,r^{2} a_{k +1}+a d a_{k +1}+a \gamma a_{k +1}-a_{k} \alpha \beta -a_{k} \alpha k +\alpha k a_{k +1}-\alpha r a_{k}+\alpha r a_{k +1}-a_{k} \beta k +\beta k a_{k +1}-\beta r a_{k}+\beta r a_{k +1}-k^{2} a_{k}+k^{2} a_{k +1}-2 k r a_{k}+2 k r a_{k +1}-r^{2} a_{k}+r^{2} a_{k +1}-a a_{k +1}+\alpha a_{k +1}+\beta a_{k +1}+2 k a_{k +1}+q a_{k +1}+2 r a_{k +1}+a_{k +1}}{a \left (k +2+r \right ) \left (\gamma +k +1+r \right )} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =0 \\ {} & {} & a_{k +2}=\frac {a d k a_{k +1}+a \gamma k a_{k +1}+a \,k^{2} a_{k +1}+a d a_{k +1}+a \gamma a_{k +1}-a_{k} \alpha \beta -a_{k} \alpha k +\alpha k a_{k +1}-a_{k} \beta k +\beta k a_{k +1}-k^{2} a_{k}+k^{2} a_{k +1}-a a_{k +1}+\alpha a_{k +1}+\beta a_{k +1}+2 k a_{k +1}+q a_{k +1}+a_{k +1}}{a \left (k +2\right ) \left (\gamma +k +1\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =0 \\ {} & {} & \left [y \left (x \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k}, a_{k +2}=\frac {a d k a_{k +1}+a \gamma k a_{k +1}+a \,k^{2} a_{k +1}+a d a_{k +1}+a \gamma a_{k +1}-a_{k} \alpha \beta -a_{k} \alpha k +\alpha k a_{k +1}-a_{k} \beta k +\beta k a_{k +1}-k^{2} a_{k}+k^{2} a_{k +1}-a a_{k +1}+\alpha a_{k +1}+\beta a_{k +1}+2 k a_{k +1}+q a_{k +1}+a_{k +1}}{a \left (k +2\right ) \left (\gamma +k +1\right )}, -a \gamma a_{1}+q a_{0}=0\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =1-\gamma \\ {} & {} & a_{k +2}=\frac {-\left (1-\gamma \right )^{2} a_{k}+\left (1-\gamma \right )^{2} a_{k +1}+2 \left (1-\gamma \right ) a_{k +1}+a \left (1-\gamma \right )^{2} a_{k +1}-a_{k} \alpha \left (1-\gamma \right )+\alpha \left (1-\gamma \right ) a_{k +1}-a_{k} \beta \left (1-\gamma \right )+\beta \left (1-\gamma \right ) a_{k +1}-2 k \left (1-\gamma \right ) a_{k}+2 k \left (1-\gamma \right ) a_{k +1}+a d a_{k +1}+\alpha k a_{k +1}+\beta k a_{k +1}+a d \left (1-\gamma \right ) a_{k +1}+a \gamma \left (1-\gamma \right ) a_{k +1}+2 a k \left (1-\gamma \right ) a_{k +1}+a d k a_{k +1}+\alpha a_{k +1}+\beta a_{k +1}+2 k a_{k +1}+q a_{k +1}+k^{2} a_{k +1}+a \,k^{2} a_{k +1}+a \gamma a_{k +1}-a_{k} \alpha k -a_{k} \beta k -a_{k} \alpha \beta +a_{k +1}-k^{2} a_{k}-a a_{k +1}+a \gamma k a_{k +1}}{a \left (k +3-\gamma \right ) \left (k +2\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =1-\gamma \\ {} & {} & \left [y \left (x \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{-\gamma +k +1}, a_{k +2}=\frac {-\left (1-\gamma \right )^{2} a_{k}+\left (1-\gamma \right )^{2} a_{k +1}+2 \left (1-\gamma \right ) a_{k +1}+a \left (1-\gamma \right )^{2} a_{k +1}-a_{k} \alpha \left (1-\gamma \right )+\alpha \left (1-\gamma \right ) a_{k +1}-a_{k} \beta \left (1-\gamma \right )+\beta \left (1-\gamma \right ) a_{k +1}-2 k \left (1-\gamma \right ) a_{k}+2 k \left (1-\gamma \right ) a_{k +1}+a d a_{k +1}+\alpha k a_{k +1}+\beta k a_{k +1}+a d \left (1-\gamma \right ) a_{k +1}+a \gamma \left (1-\gamma \right ) a_{k +1}+2 a k \left (1-\gamma \right ) a_{k +1}+a d k a_{k +1}+\alpha a_{k +1}+\beta a_{k +1}+2 k a_{k +1}+q a_{k +1}+k^{2} a_{k +1}+a \,k^{2} a_{k +1}+a \gamma a_{k +1}-a_{k} \alpha k -a_{k} \beta k -a_{k} \alpha \beta +a_{k +1}-k^{2} a_{k}-a a_{k +1}+a \gamma k a_{k +1}}{a \left (k +3-\gamma \right ) \left (k +2\right )}, -a a_{1} \left (-\gamma +2\right )+a_{0} \left (a d \left (1-\gamma \right )+a \gamma \left (1-\gamma \right )+a \left (1-\gamma \right )^{2}-2 a \left (1-\gamma \right )+\alpha \left (1-\gamma \right )+\beta \left (1-\gamma \right )+\left (1-\gamma \right )^{2}+q \right )=0\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y \left (x \right )=\left (\moverset {\infty }{\munderset {k =0}{\sum }}b_{k} x^{k}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}c_{k} x^{-\gamma +k +1}\right ), b_{k +2}=\frac {a d k b_{k +1}+a \gamma k b_{k +1}+a \,k^{2} b_{k +1}+a d b_{k +1}+a \gamma b_{k +1}-\alpha \beta b_{k}-\alpha k b_{k}+\alpha k b_{k +1}-\beta k b_{k}+\beta k b_{k +1}-k^{2} b_{k}+k^{2} b_{k +1}-a b_{k +1}+\alpha b_{k +1}+\beta b_{k +1}+2 k b_{k +1}+q b_{k +1}+b_{k +1}}{a \left (k +2\right ) \left (\gamma +k +1\right )}, -a \gamma b_{1}+q b_{0}=0, c_{k +2}=\frac {c_{k +1}-\left (1-\gamma \right )^{2} c_{k}+\left (1-\gamma \right )^{2} c_{k +1}+2 \left (1-\gamma \right ) c_{k +1}+\alpha c_{k +1}+\beta c_{k +1}+2 k c_{k +1}+q c_{k +1}+k^{2} c_{k +1}-k^{2} c_{k}-a c_{k +1}+a \left (1-\gamma \right )^{2} c_{k +1}-c_{k} \alpha \left (1-\gamma \right )+\alpha \left (1-\gamma \right ) c_{k +1}-c_{k} \beta \left (1-\gamma \right )+\beta \left (1-\gamma \right ) c_{k +1}-2 k \left (1-\gamma \right ) c_{k}+2 k \left (1-\gamma \right ) c_{k +1}+a d c_{k +1}+\alpha k c_{k +1}+\beta k c_{k +1}+a \,k^{2} c_{k +1}+a \gamma c_{k +1}-c_{k} \alpha k -c_{k} \beta k -c_{k} \alpha \beta +a d \left (1-\gamma \right ) c_{k +1}+a \gamma \left (1-\gamma \right ) c_{k +1}+2 a k \left (1-\gamma \right ) c_{k +1}+a d k c_{k +1}+a \gamma k c_{k +1}}{a \left (k +3-\gamma \right ) \left (k +2\right )}, -a c_{1} \left (-\gamma +2\right )+c_{0} \left (a d \left (1-\gamma \right )+a \gamma \left (1-\gamma \right )+a \left (1-\gamma \right )^{2}-2 a \left (1-\gamma \right )+\alpha \left (1-\gamma \right )+\beta \left (1-\gamma \right )+\left (1-\gamma \right )^{2}+q \right )=0\right ] \end {array} \]

2.32.22.2 ✓ Mathematica. Time used: 1.42 (sec). Leaf size: 85

ode=x*(x-1)*(x-a)*D[y[x],{x,2}]+((\[Alpha]+\[Beta]+1)*x^2-(\[Alpha]+\[Beta]+1+a*(\[Gamma]+d)-a)*x+a*\[Gamma])*D[y[x],x]+(\[Alpha]*\[Beta]*x-q)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_2 x^{1-\gamma } \text {HeunG}\left [a,q-(\gamma -1) (a (d-1)+\alpha +\beta -\gamma +1),\alpha -\gamma +1,\beta -\gamma +1,2-\gamma ,\frac {a (d-1)}{a-1},x\right ]+c_1 \text {HeunG}\left [a,q,\alpha ,\beta ,\gamma ,\frac {a (d-1)}{a-1},x\right ] \end{align*}

2.32.22.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
Alpha = symbols("Alpha") 
BETA = symbols("BETA") 
Gamma = symbols("Gamma") 
a = symbols("a") 
d = symbols("d") 
q = symbols("q") 
y = Function("y") 
ode = Eq(x*(-a + x)*(x - 1)*Derivative(y(x), (x, 2)) + (Alpha*BETA*x - q)*y(x) + (Gamma*a + x**2*(Alpha + BETA + 1) - x*(Alpha + BETA + a*(Gamma + d) - a + 1))*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None
 

Python version: 3.12.3 (main, Aug 14 2025, 17:47:21) [GCC 13.3.0] 
Sympy version 1.14.0
 

classify_ode(ode,func=y(x)) 
 
('factorable', '2nd_power_series_regular')