54.3.103 problem 1117

Internal problem ID [12398]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1117
Date solved : Friday, October 03, 2025 at 03:18:55 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.056 (sec). Leaf size: 82

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

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

✓ Mathematica. Time used: 1.078 (sec). Leaf size: 98

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

\begin{align*} y(x)&\to \left (c_1 \operatorname {HypergeometricU}\left (\frac {a^2+b a+a-b}{a-b},a+b+2,(a-b) x\right )+c_2 L_{-\frac {a^2+b a+a-b}{a-b}}^{a+b+1}((a-b) x)\right ) \exp \left (\int _1^x\frac {a+b+b K[1]+1}{K[1]}dK[1]\right ) \end{align*}

✗ Sympy

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

ValueError : Expected Expr or iterable but got None