54.3.377 problem 1394

Internal problem ID [12672]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1394
Date solved : Wednesday, October 01, 2025 at 02:19:26 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 79

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

\[ y = \sqrt {\frac {a x +b}{x}}\, \left (\left (\frac {x}{a x +b}\right )^{-\frac {\sqrt {\frac {b^{2}-4 c}{a^{2}}}\, a}{2 b}} c_2 +\left (\frac {x}{a x +b}\right )^{\frac {\sqrt {\frac {b^{2}-4 c}{a^{2}}}\, a}{2 b}} c_1 \right ) \]

✓ Mathematica. Time used: 2.468 (sec). Leaf size: 117

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

\begin{align*} y(x)&\to \frac {\exp \left (\int _1^x\frac {\sqrt {1-\frac {4 c}{b^2}} b+b+2 a K[1]}{2 K[1] (b+a K[1])}dK[1]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[2]}\frac {\sqrt {1-\frac {4 c}{b^2}} b+b+2 a K[1]}{2 K[1] (b+a K[1])}dK[1]\right )dK[2]+c_1\right )}{x} \end{align*}

✗ Sympy

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

ValueError : Expected Expr or iterable but got None