55.30.26 problem 135

Internal problem ID [13908]
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 : 135
Date solved : Friday, October 03, 2025 at 06:55:35 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.128 (sec). Leaf size: 79

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

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

✓ Mathematica. Time used: 0.072 (sec). Leaf size: 157

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

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

✗ Sympy

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

ValueError : Expected Expr or iterable but got None