71.1.38 problem 55 (page 96)

Internal problem ID [19214]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 55 (page 96)
Date solved : Thursday, October 02, 2025 at 03:44:37 PM
CAS classification : [_exact, _rational]

\begin{align*} \frac {1}{x}-\frac {y^{2}}{\left (x -y\right )^{2}}+\left (\frac {x^{2}}{\left (x -y\right )^{2}}-\frac {1}{y}\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.035 (sec). Leaf size: 38

ode:=1/x-y(x)^2/(x-y(x))^2+(x^2/(x-y(x))^2-1/y(x))*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{\operatorname {RootOf}\left (\ln \left (x \right ) {\mathrm e}^{\textit {\_Z}}+c_1 \,{\mathrm e}^{\textit {\_Z}}-\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{\textit {\_Z}} x -\ln \left (x \right ) x -c_1 x +\textit {\_Z} x \right )} \]

✓ Mathematica. Time used: 0.115 (sec). Leaf size: 164

ode=(1/y[x]*Sin[x/y[x]]-y[x]/x^2*Cos[y[x]/x]+1 )+( 1/x*Cos[y[x]/x]-x/y[x]^2*Sin[x/y[x]]+1/y[x]^2 ) *D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^x\left (\frac {\cos \left (\frac {y(x)}{K[1]}\right ) y(x)}{K[1]^2}-\frac {\sin \left (\frac {K[1]}{y(x)}\right )+y(x)}{y(x)}\right )dK[1]+\int _1^{y(x)}\left (\frac {x \sin \left (\frac {x}{K[2]}\right )-1}{K[2]^2}-\frac {\cos \left (\frac {K[2]}{x}\right )+x \int _1^x\left (\frac {\cos \left (\frac {K[2]}{K[1]}\right )}{K[1]^2}+\frac {K[2]+\sin \left (\frac {K[1]}{K[2]}\right )}{K[2]^2}-\frac {K[2] \sin \left (\frac {K[2]}{K[1]}\right )}{K[1]^3}-\frac {1-\frac {\cos \left (\frac {K[1]}{K[2]}\right ) K[1]}{K[2]^2}}{K[2]}\right )dK[1]}{x}\right )dK[2]=c_1,y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x**2/(x - y(x))**2 - 1/y(x))*Derivative(y(x), x) - y(x)**2/(x - y(x))**2 + 1/x,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out