62.12.26 problem Ex 27

Internal problem ID [12872]
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 : Tuesday, January 28, 2025 at 04:29:48 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*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 44

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

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

✓ Solution by Mathematica

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

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

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