2.30.22 Problem 131

Internal problem ID [13792]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-4
Problem number : 131
Date solved : Friday, December 19, 2025 at 12:36:30 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

2.30.22.1 ✓ Maple. Time used: 0.041 (sec). Leaf size: 75

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

\[ y = x^{-\frac {\lambda }{2}} \left (\operatorname {WhittakerM}\left (-\frac {i b}{2 \sqrt {a}}, \frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}, 2 i \sqrt {a}\, x \right ) c_1 +\operatorname {WhittakerW}\left (-\frac {i b}{2 \sqrt {a}}, \frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}, 2 i \sqrt {a}\, x \right ) c_2 \right ) \]

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 
   -> Whittaker 
      -> hyper3: Equivalence to 1F1 under a power @ Moebius 
      <- hyper3 successful: received ODE is equivalent to the 1F1 ODE 
   <- Whittaker successful 
<- special function solution successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} \left (\frac {d^{2}}{d x^{2}}y \left (x \right )\right )+\lambda x \left (\frac {d}{d x}y \left (x \right )\right )+\left (a \,x^{2}+b x +c \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 (a \,x^{2}+b x +c \right ) y \left (x \right )}{x^{2}}-\frac {\lambda \left (\frac {d}{d x}y \left (x \right )\right )}{x} \\ \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 {\lambda \left (\frac {d}{d x}y \left (x \right )\right )}{x}+\frac {\left (a \,x^{2}+b x +c \right ) y \left (x \right )}{x^{2}}=0 \\ \square & {} & \textrm {Check to see if}\hspace {3pt} x_{0}=0\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & \circ & \textrm {Define functions}\hspace {3pt} \\ {} & {} & \left [P_{2}\left (x \right )=\frac {\lambda }{x}, P_{3}\left (x \right )=\frac {a \,x^{2}+b x +c}{x^{2}}\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}}}=\lambda \\ {} & \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}}}=c \\ {} & \circ & x =0\textrm {is a regular singular point}\hspace {3pt} \\ & {} & \textrm {Check to see if}\hspace {3pt} x_{0}=0\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & {} & x_{0}=0 \\ \bullet & {} & \textrm {Multiply by denominators}\hspace {3pt} \\ {} & {} & x^{2} \left (\frac {d^{2}}{d x^{2}}y \left (x \right )\right )+\lambda x \left (\frac {d}{d x}y \left (x \right )\right )+\left (a \,x^{2}+b x +c \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..2 \\ {} & {} & 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 \cdot \left (\frac {d}{d x}y \left (x \right )\right )\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & x \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} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{2}\cdot \left (\frac {d^{2}}{d x^{2}}y \left (x \right )\right )\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & x^{2}\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} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & a_{0} \left (\lambda r +r^{2}+c -r \right ) x^{r}+\left (\left (\lambda r +r^{2}+c +\lambda +r \right ) a_{1}+a_{0} b \right ) x^{1+r}+\left (\moverset {\infty }{\munderset {k =2}{\sum }}\left (a_{k} \left (k^{2}+\lambda k +2 k r +\lambda r +r^{2}+c -k -r \right )+a_{k -1} b +a_{k -2} a \right ) x^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & \lambda r +r^{2}+c -r =0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{-\frac {\lambda }{2}+\frac {1}{2}-\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}, -\frac {\lambda }{2}+\frac {1}{2}+\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}\right \} \\ \bullet & {} & \textrm {Each term must be 0}\hspace {3pt} \\ {} & {} & \left (\lambda r +r^{2}+c +\lambda +r \right ) a_{1}+a_{0} b =0 \\ \bullet & {} & \textrm {Solve for the dependent coefficient(s)}\hspace {3pt} \\ {} & {} & a_{1}=-\frac {a_{0} b}{\lambda r +r^{2}+c +\lambda +r} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & \left (k^{2}+\left (2 r +\lambda -1\right ) k +r^{2}+\left (\lambda -1\right ) r +c \right ) a_{k}+a_{k -2} a +a_{k -1} b =0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +2 \\ {} & {} & \left (\left (k +2\right )^{2}+\left (2 r +\lambda -1\right ) \left (k +2\right )+r^{2}+\left (\lambda -1\right ) r +c \right ) a_{k +2}+a_{k} a +a_{k +1} b =0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +2}=-\frac {a_{k} a +a_{k +1} b}{k^{2}+\lambda k +2 k r +\lambda r +r^{2}+c +3 k +2 \lambda +3 r +2} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =-\frac {\lambda }{2}+\frac {1}{2}-\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2} \\ {} & {} & a_{k +2}=-\frac {a_{k} a +a_{k +1} b}{k^{2}+\lambda k +2 k \left (-\frac {\lambda }{2}+\frac {1}{2}-\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}\right )+\lambda \left (-\frac {\lambda }{2}+\frac {1}{2}-\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}\right )+{\left (-\frac {\lambda }{2}+\frac {1}{2}-\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}\right )}^{2}+c +3 k +\frac {\lambda }{2}+\frac {7}{2}-\frac {3 \sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =-\frac {\lambda }{2}+\frac {1}{2}-\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2} \\ {} & {} & \left [y \left (x \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k -\frac {\lambda }{2}+\frac {1}{2}-\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}}, a_{k +2}=-\frac {a_{k} a +a_{k +1} b}{k^{2}+\lambda k +2 k \left (-\frac {\lambda }{2}+\frac {1}{2}-\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}\right )+\lambda \left (-\frac {\lambda }{2}+\frac {1}{2}-\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}\right )+{\left (-\frac {\lambda }{2}+\frac {1}{2}-\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}\right )}^{2}+c +3 k +\frac {\lambda }{2}+\frac {7}{2}-\frac {3 \sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}}, a_{1}=-\frac {a_{0} b}{\lambda \left (-\frac {\lambda }{2}+\frac {1}{2}-\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}\right )+{\left (-\frac {\lambda }{2}+\frac {1}{2}-\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}\right )}^{2}+c +\frac {\lambda }{2}+\frac {1}{2}-\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}}\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =-\frac {\lambda }{2}+\frac {1}{2}+\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2} \\ {} & {} & a_{k +2}=-\frac {a_{k} a +a_{k +1} b}{k^{2}+\lambda k +2 k \left (-\frac {\lambda }{2}+\frac {1}{2}+\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}\right )+\lambda \left (-\frac {\lambda }{2}+\frac {1}{2}+\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}\right )+{\left (-\frac {\lambda }{2}+\frac {1}{2}+\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}\right )}^{2}+c +3 k +\frac {\lambda }{2}+\frac {7}{2}+\frac {3 \sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =-\frac {\lambda }{2}+\frac {1}{2}+\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2} \\ {} & {} & \left [y \left (x \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k -\frac {\lambda }{2}+\frac {1}{2}+\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}}, a_{k +2}=-\frac {a_{k} a +a_{k +1} b}{k^{2}+\lambda k +2 k \left (-\frac {\lambda }{2}+\frac {1}{2}+\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}\right )+\lambda \left (-\frac {\lambda }{2}+\frac {1}{2}+\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}\right )+{\left (-\frac {\lambda }{2}+\frac {1}{2}+\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}\right )}^{2}+c +3 k +\frac {\lambda }{2}+\frac {7}{2}+\frac {3 \sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}}, a_{1}=-\frac {a_{0} b}{\lambda \left (-\frac {\lambda }{2}+\frac {1}{2}+\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}\right )+{\left (-\frac {\lambda }{2}+\frac {1}{2}+\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}\right )}^{2}+c +\frac {\lambda }{2}+\frac {1}{2}+\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}}\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y \left (x \right )=\left (\moverset {\infty }{\munderset {k =0}{\sum }}d_{k} x^{k -\frac {\lambda }{2}+\frac {1}{2}-\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}e_{k} x^{k -\frac {\lambda }{2}+\frac {1}{2}+\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}}\right ), d_{k +2}=-\frac {a d_{k}+b d_{k +1}}{k^{2}+\lambda k +2 k \left (-\frac {\lambda }{2}+\frac {1}{2}-\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}\right )+\lambda \left (-\frac {\lambda }{2}+\frac {1}{2}-\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}\right )+{\left (-\frac {\lambda }{2}+\frac {1}{2}-\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}\right )}^{2}+c +3 k +\frac {\lambda }{2}+\frac {7}{2}-\frac {3 \sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}}, d_{1}=-\frac {d_{0} b}{\lambda \left (-\frac {\lambda }{2}+\frac {1}{2}-\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}\right )+{\left (-\frac {\lambda }{2}+\frac {1}{2}-\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}\right )}^{2}+c +\frac {\lambda }{2}+\frac {1}{2}-\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}}, e_{k +2}=-\frac {a e_{k}+b e_{k +1}}{k^{2}+\lambda k +2 k \left (-\frac {\lambda }{2}+\frac {1}{2}+\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}\right )+\lambda \left (-\frac {\lambda }{2}+\frac {1}{2}+\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}\right )+{\left (-\frac {\lambda }{2}+\frac {1}{2}+\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}\right )}^{2}+c +3 k +\frac {\lambda }{2}+\frac {7}{2}+\frac {3 \sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}}, e_{1}=-\frac {e_{0} b}{\lambda \left (-\frac {\lambda }{2}+\frac {1}{2}+\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}\right )+{\left (-\frac {\lambda }{2}+\frac {1}{2}+\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}\right )}^{2}+c +\frac {\lambda }{2}+\frac {1}{2}+\frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}}\right ] \end {array} \]

2.30.22.2 ✓ Mathematica. Time used: 0.112 (sec). Leaf size: 159

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

\begin{align*} y(x)&\to e^{-i \sqrt {a} x} x^{\frac {1}{2} \left (\sqrt {(\lambda -1)^2-4 c}-\lambda +1\right )} \left (c_1 \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {i b}{\sqrt {a}}+\sqrt {(\lambda -1)^2-4 c}+1\right ),\sqrt {(\lambda -1)^2-4 c}+1,2 i \sqrt {a} x\right )+c_2 L_{\frac {1}{2} \left (-\frac {i b}{\sqrt {a}}-\sqrt {(\lambda -1)^2-4 c}-1\right )}^{\sqrt {(\lambda -1)^2-4 c}}\left (2 i \sqrt {a} x\right )\right ) \end{align*}

2.30.22.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
lambda_ = symbols("lambda_") 
y = Function("y") 
ode = Eq(lambda_*x*Derivative(y(x), x) + x**2*Derivative(y(x), (x, 2)) + (a*x**2 + b*x + c)*y(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')