2.28.21 Problem 31

Internal problem ID [13692]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-2
Problem number : 31
Date solved : Friday, December 19, 2025 at 10:40:32 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

2.28.21.1 ✓ Maple. Time used: 0.010 (sec). Leaf size: 254

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

\[ y = {\mathrm e}^{-\frac {\left (\left (a x +2 b \right ) \sqrt {a^{2}-4 \alpha }+x \left (a^{2}-4 \alpha \right )+2 a b -4 \beta \right ) x}{4 \sqrt {a^{2}-4 \alpha }}} \left (c_2 \left (x \,a^{2}+a b -4 \alpha x -2 \beta \right ) \operatorname {hypergeom}\left (\left [\frac {3 \left (a^{2}-4 \alpha \right )^{{3}/{2}}+a^{3}-2 a^{2} \gamma +2 \left (\beta b -2 \alpha \right ) a +2 \left (-b^{2}+4 \gamma \right ) \alpha -2 \beta ^{2}}{4 \left (a^{2}-4 \alpha \right )^{{3}/{2}}}\right ], \left [\frac {3}{2}\right ], \frac {\left (x \,a^{2}+a b -4 \alpha x -2 \beta \right )^{2}}{2 \left (a^{2}-4 \alpha \right )^{{3}/{2}}}\right )+c_1 \operatorname {hypergeom}\left (\left [\frac {\left (a^{2}-4 \alpha \right )^{{3}/{2}}+a^{3}-2 a^{2} \gamma +\left (2 \beta b -4 \alpha \right ) a +\left (-2 b^{2}+8 \gamma \right ) \alpha -2 \beta ^{2}}{4 \left (a^{2}-4 \alpha \right )^{{3}/{2}}}\right ], \left [\frac {1}{2}\right ], \frac {\left (x \,a^{2}+a b -4 \alpha x -2 \beta \right )^{2}}{2 \left (a^{2}-4 \alpha \right )^{{3}/{2}}}\right )\right ) \]

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 
      <- hyper3 successful: indirect Equivalence to 0F1 under ``^ @ Moebius`` i\ 
s resolved 
   <- hypergeometric successful 
<- special function solution successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d^{2}}{d x^{2}}y \left (x \right )+\left (a x +b \right ) \left (\frac {d}{d x}y \left (x \right )\right )+\left (\alpha \,x^{2}+\beta x +\gamma \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 {Assume series solution for}\hspace {3pt} y \left (x \right ) \\ {} & {} & y \left (x \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k} \\ \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..2 \\ {} & {} & x^{m}\cdot y \left (x \right )=\moverset {\infty }{\munderset {k =\max \left (0, -m \right )}{\sum }}a_{k} x^{k +m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k -m \\ {} & {} & x^{m}\cdot y \left (x \right )=\moverset {\infty }{\munderset {k =\max \left (0, -m \right )+m}{\sum }}a_{k -m} x^{k} \\ {} & \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..1 \\ {} & {} & x^{m}\cdot \left (\frac {d}{d x}y \left (x \right )\right )=\moverset {\infty }{\munderset {k =\max \left (0, 1-m \right )}{\sum }}a_{k} k \,x^{k -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 =\max \left (0, 1-m \right )+m -1}{\sum }}a_{k +1-m} \left (k +1-m \right ) x^{k} \\ {} & \circ & \textrm {Convert}\hspace {3pt} \frac {d^{2}}{d x^{2}}y \left (x \right )\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & \frac {d^{2}}{d x^{2}}y \left (x \right )=\moverset {\infty }{\munderset {k =2}{\sum }}a_{k} k \left (k -1\right ) x^{k -2} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +2 \\ {} & {} & \frac {d^{2}}{d x^{2}}y \left (x \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k +2} \left (k +2\right ) \left (k +1\right ) x^{k} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & a_{1} b +a_{0} \gamma +2 a_{2}+\left (6 a_{3}+2 a_{2} b +a_{1} \left (a +\gamma \right )+a_{0} \beta \right ) x +\left (\moverset {\infty }{\munderset {k =2}{\sum }}\left (a_{k +2} \left (k +2\right ) \left (k +1\right )+a_{k +1} \left (k +1\right ) b +a_{k} \left (a k +\gamma \right )+a_{k -1} \beta +a_{k -2} \alpha \right ) x^{k}\right )=0 \\ \bullet & {} & \textrm {The coefficients of each power of}\hspace {3pt} x \hspace {3pt}\textrm {must be 0}\hspace {3pt} \\ {} & {} & \left [2 a_{2}+a_{1} b +a_{0} \gamma =0, 6 a_{3}+2 a_{2} b +a_{1} \left (a +\gamma \right )+a_{0} \beta =0\right ] \\ \bullet & {} & \textrm {Solve for the dependent coefficient(s)}\hspace {3pt} \\ {} & {} & \left \{a_{2}=-\frac {a_{1} b}{2}-\frac {a_{0} \gamma }{2}, a_{3}=\frac {1}{6} a_{1} b^{2}+\frac {1}{6} a_{0} b \gamma -\frac {1}{6} a_{1} a -\frac {1}{6} a_{0} \beta -\frac {1}{6} a_{1} \gamma \right \} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & k^{2} a_{k +2}+\left (a a_{k}+a_{k +1} b +3 a_{k +2}\right ) k +a_{k +1} b +a_{k -1} \beta +a_{k -2} \alpha +a_{k} \gamma +2 a_{k +2}=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +2 \\ {} & {} & \left (k +2\right )^{2} a_{k +4}+\left (a a_{k +2}+a_{k +3} b +3 a_{k +4}\right ) \left (k +2\right )+a_{k +3} b +a_{k +1} \beta +a_{k} \alpha +a_{k +2} \gamma +2 a_{k +4}=0 \\ \bullet & {} & \textrm {Recursion relation that defines the series solution to the ODE}\hspace {3pt} \\ {} & {} & \left [y \left (x \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k}, a_{k +4}=-\frac {a k a_{k +2}+b k a_{k +3}+2 a a_{k +2}+a_{k} \alpha +3 a_{k +3} b +a_{k +1} \beta +a_{k +2} \gamma }{k^{2}+7 k +12}, a_{2}=-\frac {a_{1} b}{2}-\frac {a_{0} \gamma }{2}, a_{3}=\frac {1}{6} a_{1} b^{2}+\frac {1}{6} a_{0} b \gamma -\frac {1}{6} a_{1} a -\frac {1}{6} a_{0} \beta -\frac {1}{6} a_{1} \gamma \right ] \end {array} \]

2.28.21.2 ✓ Mathematica. Time used: 0.193 (sec). Leaf size: 307

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

\begin{align*} y(x)&\to \exp \left (-\frac {x \left (2 b \sqrt {a^2-4 \alpha }+a \left (x \sqrt {a^2-4 \alpha }+2 b\right )+a^2 x-4 (\beta +\alpha x)\right )}{4 \sqrt {a^2-4 \alpha }}\right ) \left (c_1 \operatorname {HermiteH}\left (\frac {-a^3-\left (\sqrt {a^2-4 \alpha }-2 \gamma \right ) a^2+(4 \alpha -2 b \beta ) a+2 \left (\alpha b^2+\beta ^2+2 \sqrt {a^2-4 \alpha } \alpha -4 \alpha \gamma \right )}{2 \left (a^2-4 \alpha \right )^{3/2}},\frac {x a^2+b a-2 (2 x \alpha +\beta )}{\sqrt {2} \left (a^2-4 \alpha \right )^{3/4}}\right )+c_2 \operatorname {Hypergeometric1F1}\left (\frac {a^3+\left (\sqrt {a^2-4 \alpha }-2 \gamma \right ) a^2+(2 b \beta -4 \alpha ) a-2 \left (\alpha b^2+\beta ^2+2 \sqrt {a^2-4 \alpha } \alpha -4 \alpha \gamma \right )}{4 \left (a^2-4 \alpha \right )^{3/2}},\frac {1}{2},\frac {\left (x a^2+b a-2 (2 x \alpha +\beta )\right )^2}{2 \left (a^2-4 \alpha \right )^{3/2}}\right )\right ) \end{align*}

2.28.21.3 ✗ Sympy

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

False