60.3.159 problem 1173

Internal problem ID [11155]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1173
Date solved : Thursday, March 13, 2025 at 08:24:51 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+2 \left (x +a \right ) y^{\prime }-b \left (b -1\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.186 (sec). Leaf size: 37

ode:=x^2*diff(diff(y(x),x),x)+2*(x+a)*diff(y(x),x)-b*(b-1)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{\frac {a}{x}} \left (\operatorname {BesselI}\left (b -\frac {1}{2}, \frac {a}{x}\right ) c_{1} +\operatorname {BesselK}\left (b -\frac {1}{2}, \frac {a}{x}\right ) c_{2} \right )}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.194 (sec). Leaf size: 74

ode=(1 - b)*b*y[x] + 2*(a + x)*D[y[x],x] + x^2*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(-b*(b - 1)*y(x) + x**2*Derivative(y(x), (x, 2)) + (2*a + 2*x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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