62.12.26 problem Ex 27

Internal problem ID [12801]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter 2, differential equations of the first order and the first degree. Article 19. Summary. Page 29
Problem number : Ex 27
Date solved : Monday, March 31, 2025 at 07:09:06 AM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 44

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

\[ y = \operatorname {RootOf}\left (\int _{}^{\textit {\_Z}}\frac {\textit {\_a} -1}{\textit {\_a} \left ({\mathrm e}^{-\frac {1}{\textit {\_a}}}+{\mathrm e}^{\frac {\left (\textit {\_a} -1\right ) \left (\textit {\_a} +1\right )}{\textit {\_a}}}+\textit {\_a} -1\right )}d \textit {\_a} +\ln \left (x \right )+c_1 \right ) x \]

✓ Mathematica. Time used: 0.338 (sec). Leaf size: 63

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

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

✓ Sympy. Time used: 27.432 (sec). Leaf size: 58

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

\[ y{\left (x \right )} = C_{1} e^{- \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {e^{\frac {1}{u_{1}}}}{u_{1} e^{\frac {1}{u_{1}}} - u_{1} e^{u_{1}} + u_{1} + e^{u_{1}}}\, du_{1} - \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {1}{u_{1} e^{\frac {1}{u_{1}}} - u_{1} e^{u_{1}} + u_{1} + e^{u_{1}}}\, du_{1}} \]