60.3.147 problem 1161

Internal problem ID [11143]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1161
Date solved : Sunday, March 30, 2025 at 07:44:00 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 31

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

\[ y = c_1 \operatorname {BesselI}\left (2 \sqrt {a}, 2 \sqrt {x}\right )+c_2 \operatorname {BesselK}\left (2 \sqrt {a}, 2 \sqrt {x}\right ) \]

✓ Mathematica. Time used: 0.085 (sec). Leaf size: 78

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

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

✓ Sympy. Time used: 0.205 (sec). Leaf size: 36

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

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