54.3.323 problem 1340

Internal problem ID [12618]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1340
Date solved : Wednesday, October 01, 2025 at 02:16:00 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=\frac {2 \left (a x +2 b \right ) y^{\prime }}{x \left (a x +b \right )}-\frac {\left (2 a x +6 b \right ) y}{\left (a x +b \right ) x^{2}} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 20

ode:=diff(diff(y(x),x),x) = 2/x*(a*x+2*b)/(a*x+b)*diff(y(x),x)-(2*a*x+6*b)/(a*x+b)/x^2*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {x^{2} \left (c_2 x +c_1 \right )}{a x +b} \]

✓ Mathematica. Time used: 0.114 (sec). Leaf size: 44

ode=D[y[x],{x,2}] == -(((6*b + 2*a*x)*y[x])/(x^2*(b + a*x))) + (2*(2*b + a*x)*D[y[x],x])/(x*(b + a*x)); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to (c_2 x+c_1) \exp \left (-\frac {1}{2} \int _1^x\left (\frac {2 a}{b+a K[1]}-\frac {4}{K[1]}\right )dK[1]\right ) \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)) - (2*a*x + 4*b)*Derivative(y(x), x)/(x*(a*x + b)) + (2*a*x + 6*b)*y(x)/(x**2*(a*x + b)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False