61.29.25 problem 134

Internal problem ID [12555]
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 : 134
Date solved : Monday, March 31, 2025 at 05:38:51 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.068 (sec). Leaf size: 114

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

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

✓ Mathematica. Time used: 0.315 (sec). Leaf size: 243

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

\[ y(x)\to -i^{-\sqrt {a^2-2 a-4 c+1}+a+1} b^{\frac {1}{2} \left (-\sqrt {a^2-2 a-4 c+1}+a-1\right )} \left (\frac {1}{x}\right )^{\frac {1}{2} \left (-\sqrt {a^2-2 a-4 c+1}+a-1\right )} \left (c_2 i^{2 \sqrt {a^2-2 a-4 c+1}} b^{\sqrt {a^2-2 a-4 c+1}} \left (\frac {1}{x}\right )^{\sqrt {a^2-2 a-4 c+1}} \operatorname {Hypergeometric1F1}\left (\frac {1}{2} \left (a+\sqrt {a^2-2 a-4 c+1}-1\right ),\sqrt {a^2-2 a-4 c+1}+1,\frac {b}{x}\right )+c_1 \operatorname {Hypergeometric1F1}\left (\frac {1}{2} \left (a-\sqrt {a^2-2 a-4 c+1}-1\right ),1-\sqrt {a^2-2 a-4 c+1},\frac {b}{x}\right )\right ) \]

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (-c*y(x) - x**2*Derivative(y(x), (x, 2)))/(a*x + b) cannot be solved by the factorable group method