54.3.329 problem 1346

Internal problem ID [12624]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1346
Date solved : Wednesday, October 01, 2025 at 02:18:29 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=\frac {\left (a +b \right ) y^{\prime }}{x^{2}}-\frac {\left (\left (a +b \right ) x +a b \right ) y}{x^{4}} \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 25

ode:=diff(diff(y(x),x),x) = 1/x^2*(a+b)*diff(y(x),x)-(x*(a+b)+a*b)/x^4*y(x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.134 (sec). Leaf size: 41

ode=D[y[x],{x,2}] == -(((a*b + (a + b)*x)*y[x])/x^4) + ((a + b)*D[y[x],x])/x^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 2)) - (a + b)*Derivative(y(x), x)/x**2 + (a*b + x*(a + b))*y(x)/x**4,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (a*b*y(x) + a*x*y(x) + b*x*y(x) + x**4*Derivative(y(x), (x, 2)))/(x**2*(a + b)) cannot be solved by the factorable group method