75.8.11 problem 209

Internal problem ID [16827]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 8. First order not solved for the derivative. Exercises page 67
Problem number : 209
Date solved : Tuesday, January 28, 2025 at 09:34:05 AM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }&={\mathrm e}^{\frac {y^{\prime }}{y}} \end{align*}

✓ Solution by Maple

Time used: 0.303 (sec). Leaf size: 25

dsolve(diff(y(x),x)=exp(diff(y(x),x)/y(x)),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.052 (sec). Leaf size: 35

DSolve[D[y[x],x]==Exp[D[y[x],x]/y[x]],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] W\left (-\frac {1}{K[1]}\right )}dK[1]\&\right ][-x+c_1] \]