75.8.10 problem 208

Internal problem ID [16826]
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 : 208
Date solved : Tuesday, January 28, 2025 at 09:34:04 AM
CAS classification : [_quadrature]

\begin{align*} y&={y^{\prime }}^{2} {\mathrm e}^{y^{\prime }} \end{align*}

✓ Solution by Maple

Time used: 0.265 (sec). Leaf size: 38

dsolve(y(x)=diff(y(x),x)^2*exp(diff(y(x),x)),y(x), singsol=all)
 

\begin{align*} y &= 0 \\ y &= \frac {\left (x -c_{1} \right ) \left (\operatorname {LambertW}\left (\left (x -c_{1} \right ) {\mathrm e}\right )-1\right )^{2}}{\operatorname {LambertW}\left (\left (x -c_{1} \right ) {\mathrm e}\right )} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.313 (sec). Leaf size: 102

DSolve[y[x]==D[y[x],x]^2*Exp[D[y[x],x]],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \text {InverseFunction}\left [\frac {\text {$\#$1}}{W\left (-\frac {\sqrt {\text {$\#$1}}}{2}\right )}+\frac {\text {$\#$1}}{2 W\left (-\frac {\sqrt {\text {$\#$1}}}{2}\right )^2}\&\right ][2 x+c_1] \\ y(x)\to \text {InverseFunction}\left [\frac {\text {$\#$1}}{W\left (\frac {\sqrt {\text {$\#$1}}}{2}\right )}+\frac {\text {$\#$1}}{2 W\left (\frac {\sqrt {\text {$\#$1}}}{2}\right )^2}\&\right ][2 x+c_1] \\ y(x)\to 0 \\ \end{align*}