54.3.422 problem 1442

Internal problem ID [12717]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1442
Date solved : Wednesday, October 01, 2025 at 02:21:19 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.014 (sec). Leaf size: 33

ode:=diff(diff(y(x),x),x) = -x/f(x)*diff(y(x),x)+1/f(x)*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = x \left (\int {\mathrm e}^{-\int \frac {x^{2}+2 f \left (x \right )}{f \left (x \right ) x}d x}d x c_1 +c_2 \right ) \]

✓ Mathematica. Time used: 0.134 (sec). Leaf size: 45

ode=D[y[x],{x,2}] == y[x]/f[x] - (x*D[y[x],x])/f[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

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

TypeError : cannot determine truth value of Relational: x > _n + 2