60.3.155 problem 1169

Internal problem ID [11151]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1169
Date solved : Wednesday, March 05, 2025 at 01:43:41 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.083 (sec). Leaf size: 49

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

\[ y = \frac {c_{1} \operatorname {BesselJ}\left (\sqrt {4 b^{2}+1}, 2 \sqrt {a}\, \sqrt {x}\right )+c_{2} \operatorname {BesselY}\left (\sqrt {4 b^{2}+1}, 2 \sqrt {a}\, \sqrt {x}\right )}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.104 (sec). Leaf size: 103

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

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

✓ Sympy. Time used: 0.287 (sec). Leaf size: 58

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

\[ y{\left (x \right )} = \frac {C_{1} J_{2 \sqrt {b^{2} + \frac {1}{4}}}\left (2 \sqrt {a} \sqrt {x}\right ) + C_{2} Y_{2 \sqrt {b^{2} + \frac {1}{4}}}\left (2 \sqrt {a} \sqrt {x}\right )}{\sqrt {x}} \]