75.12.18 problem 292

Internal problem ID [16884]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 12. Miscellaneous problems. Exercises page 93
Problem number : 292
Date solved : Tuesday, January 28, 2025 at 09:39:31 AM
CAS classification : [[_homogeneous, `class A`], _exact, _dAlembert]

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

With initial conditions

\begin{align*} y \left (1\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 0.656 (sec). Leaf size: 21

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

\[ y = -\frac {x}{\operatorname {LambertW}\left (\frac {x}{-1+x -{\mathrm e}}\right )} \]

✓ Solution by Mathematica

Time used: 1.338 (sec). Leaf size: 21

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

\[ y(x)\to -\frac {x}{W\left (\frac {x}{x-e-1}\right )} \]