54.3.401 problem 1418

Internal problem ID [12696]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1418
Date solved : Wednesday, October 01, 2025 at 02:20:16 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.211 (sec). Leaf size: 47

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

\[ y = \sin \left (x \right ) \left (c_1 +c_2 \int {\mathrm e}^{-\int \frac {2 x \cos \left (x \right ) \cot \left (x \right )-3 \cos \left (x \right )+\sec \left (x \right )}{\cos \left (x \right ) x -\sin \left (x \right )}d x} \cos \left (x \right )d x \right ) \]

✓ Mathematica. Time used: 0.088 (sec). Leaf size: 15

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

\begin{align*} y(x)&\to c_1 x+c_2 \sin (x) \end{align*}

✗ Sympy

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

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