54.3.151 problem 1165

Internal problem ID [12446]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1165
Date solved : Wednesday, October 01, 2025 at 01:45:07 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 19

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

\[ y = \left (x +a \right ) c_1 +c_2 \,{\mathrm e}^{\frac {a}{x}} x \]

✓ Mathematica. Time used: 0.112 (sec). Leaf size: 31

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

\begin{align*} y(x)&\to \frac {e^2 c_2 (a+x)}{a^2}+c_1 x e^{\frac {a}{x}-1} \end{align*}

✗ Sympy

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

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