31.23 problem 204

Internal problem ID [11027]
Internal file name [OUTPUT/10284_Sunday_December_31_2023_03_59_58_PM_34416455/index.tex]

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 Equation of form \((a_3 x^3+a_2 x^2 x+a_1 x+a_0) y''+f(x)y'+g(x)y=0\)
Problem number: 204.
ODE order: 2.
ODE degree: 1.

The type(s) of ODE detected by this program : "unknown"

Maple gives the following as the ode type

[[_2nd_order, _with_linear_symmetries]]

Unable to solve or complete the solution.

\[ \boxed {x \left (x -1\right ) \left (x -a \right ) y^{\prime \prime }+\left (\left (\alpha +\beta +1\right ) x^{2}-\left (\alpha +\beta +1+a \left (\gamma +d \right )-a \right ) x +\gamma a \right ) y^{\prime }+\left (\alpha \beta x -q \right ) y=0} \]

31.23.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -\left (\frac {d}{d x}y^{\prime }\right ) x \left (x -1\right ) \left (-x +a \right )+\left (\left (\alpha +\beta +1\right ) x^{2}+\left (\left (-d -\gamma +1\right ) a -\beta -\alpha -1\right ) x +\gamma a \right ) y^{\prime }+\left (\alpha \beta x -q \right ) y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & \frac {d}{d x}y^{\prime } \\ \bullet & {} & \textrm {Isolate 2nd derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime }=\frac {\left (\alpha \beta x -q \right ) y}{x \left (x -1\right ) \left (-x +a \right )}-\frac {\left (x a d +x a \gamma -\alpha \,x^{2}-\beta \,x^{2}-\gamma a -x a +x \alpha +\beta x -x^{2}+x \right ) y^{\prime }}{x \left (x -1\right ) \left (-x +a \right )} \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y\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}{d x}y^{\prime }+\frac {\left (x a d +x a \gamma -\alpha \,x^{2}-\beta \,x^{2}-\gamma a -x a +x \alpha +\beta x -x^{2}+x \right ) y^{\prime }}{x \left (x -1\right ) \left (-x +a \right )}-\frac {\left (\alpha \beta x -q \right ) y}{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 {x a d +x a \gamma -\alpha \,x^{2}-\beta \,x^{2}-\gamma a -x a +x \alpha +\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} \\ {} & {} & \left (\frac {d}{d x}y^{\prime }\right ) x \left (x -1\right ) \left (-x +a \right )+\left (x a d +x a \gamma -\alpha \,x^{2}-\beta \,x^{2}-\gamma a -x a +x \alpha +\beta x -x^{2}+x \right ) y^{\prime }+y \left (-\alpha \beta x +q \right )=0 \\ \bullet & {} & \textrm {Assume series solution for}\hspace {3pt} y \\ {} & {} & y=\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\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =0..1 \\ {} & {} & x^{m}\cdot y=\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=\moverset {\infty }{\munderset {k =m}{\sum }}a_{k -m} x^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{m}\cdot y^{\prime }\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =0..2 \\ {} & {} & x^{m}\cdot y^{\prime }=\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 y^{\prime }=\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}{d x}y^{\prime }\right )\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =1..3 \\ {} & {} & x^{m}\cdot \left (\frac {d}{d x}y^{\prime }\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}{d x}y^{\prime }\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, -\gamma +1\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 (\beta +\alpha \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 (\beta +\alpha \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}-a_{k} \alpha r +\alpha r a_{k +1}-a_{k} \beta k +\beta k a_{k +1}-a_{k} \beta r +\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=\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 a_{1} \gamma +a_{0} q =0\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =-\gamma +1 \\ {} & {} & a_{k +2}=\frac {a d \left (-\gamma +1\right ) a_{k +1}+a \gamma \left (-\gamma +1\right ) a_{k +1}+k^{2} a_{k +1}+2 a k \left (-\gamma +1\right ) a_{k +1}-\left (-\gamma +1\right )^{2} a_{k}+\left (-\gamma +1\right )^{2} a_{k +1}+2 \left (-\gamma +1\right ) a_{k +1}+\beta k a_{k +1}+a d a_{k +1}+\alpha k a_{k +1}+a \,k^{2} a_{k +1}+a \gamma a_{k +1}-a a_{k +1}+a \gamma k a_{k +1}+a \left (-\gamma +1\right )^{2} a_{k +1}-a_{k} \alpha \left (-\gamma +1\right )+\alpha \left (-\gamma +1\right ) a_{k +1}-a_{k} \beta \left (-\gamma +1\right )+\beta \left (-\gamma +1\right ) a_{k +1}-2 k \left (-\gamma +1\right ) a_{k}+2 k \left (-\gamma +1\right ) a_{k +1}-a_{k} \alpha \beta -a_{k} \alpha k -a_{k} \beta k +a d k a_{k +1}-k^{2} a_{k}+a_{k +1}+\alpha a_{k +1}+\beta a_{k +1}+2 k a_{k +1}+q a_{k +1}}{a \left (k +3-\gamma \right ) \left (k +2\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =-\gamma +1 \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{-\gamma +k +1}, a_{k +2}=\frac {a d \left (-\gamma +1\right ) a_{k +1}+a \gamma \left (-\gamma +1\right ) a_{k +1}+k^{2} a_{k +1}+2 a k \left (-\gamma +1\right ) a_{k +1}-\left (-\gamma +1\right )^{2} a_{k}+\left (-\gamma +1\right )^{2} a_{k +1}+2 \left (-\gamma +1\right ) a_{k +1}+\beta k a_{k +1}+a d a_{k +1}+\alpha k a_{k +1}+a \,k^{2} a_{k +1}+a \gamma a_{k +1}-a a_{k +1}+a \gamma k a_{k +1}+a \left (-\gamma +1\right )^{2} a_{k +1}-a_{k} \alpha \left (-\gamma +1\right )+\alpha \left (-\gamma +1\right ) a_{k +1}-a_{k} \beta \left (-\gamma +1\right )+\beta \left (-\gamma +1\right ) a_{k +1}-2 k \left (-\gamma +1\right ) a_{k}+2 k \left (-\gamma +1\right ) a_{k +1}-a_{k} \alpha \beta -a_{k} \alpha k -a_{k} \beta k +a d k a_{k +1}-k^{2} a_{k}+a_{k +1}+\alpha a_{k +1}+\beta a_{k +1}+2 k a_{k +1}+q 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 (-\gamma +1\right )+a \gamma \left (-\gamma +1\right )+a \left (-\gamma +1\right )^{2}-2 \left (-\gamma +1\right ) a +\alpha \left (-\gamma +1\right )+\left (-\gamma +1\right ) \beta +\left (-\gamma +1\right )^{2}+q \right )=0\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y=\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}+k^{2} c_{k +1}-\left (-\gamma +1\right )^{2} c_{k}+\left (-\gamma +1\right )^{2} c_{k +1}+2 \left (-\gamma +1\right ) c_{k +1}-a c_{k +1}-k^{2} c_{k}+\alpha c_{k +1}+\beta c_{k +1}+2 k c_{k +1}+q c_{k +1}+\beta k c_{k +1}+a d c_{k +1}+\alpha k c_{k +1}+a \,k^{2} c_{k +1}+a \gamma c_{k +1}+a \left (-\gamma +1\right )^{2} c_{k +1}-c_{k} \alpha \left (-\gamma +1\right )+\alpha \left (-\gamma +1\right ) c_{k +1}-c_{k} \beta \left (-\gamma +1\right )+\beta \left (-\gamma +1\right ) c_{k +1}-2 k \left (-\gamma +1\right ) c_{k}+2 k \left (-\gamma +1\right ) c_{k +1}-c_{k} \alpha \beta -c_{k} \alpha k -c_{k} \beta k +a d \left (-\gamma +1\right ) c_{k +1}+a \gamma \left (-\gamma +1\right ) c_{k +1}+2 a k \left (-\gamma +1\right ) c_{k +1}+a \gamma k c_{k +1}+a d 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 (-\gamma +1\right )+a \gamma \left (-\gamma +1\right )+a \left (-\gamma +1\right )^{2}-2 \left (-\gamma +1\right ) a +\alpha \left (-\gamma +1\right )+\left (-\gamma +1\right ) \beta +\left (-\gamma +1\right )^{2}+q \right )=0\right ] \end {array} \]

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 @ Moebius 
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 `
 

✓ Solution by Maple

Time used: 0.578 (sec). Leaf size: 82

dsolve(x*(x-1)*(x-a)*diff(y(x),x$2)+((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,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 2.215 (sec). Leaf size: 85

DSolve[x*(x-1)*(x-a)*y''[x]+((\[Alpha]+\[Beta]+1)*x^2-(\[Alpha]+\[Beta]+1+a*(\[Gamma]+d)-a)*x+a*\[Gamma])*y'[x]+(\[Alpha]*\[Beta]*x-q)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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