Internal problem ID [15445]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Chapter 2 (Higher order ODE’s). Section 16. The method of isoclines for differential
equations of the second order. Exercises page 158
Problem number: 701.
ODE order: 2.
ODE degree: 0.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\[ \boxed {x^{\prime \prime }-x \,{\mathrm e}^{x^{\prime }}=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 33
dsolve(diff(x(t),t$2)-x(t)*exp(diff(x(t),t))=0,x(t), singsol=all)
\[ -\left (\int _{}^{x \left (t \right )}\frac {1}{\operatorname {LambertW}\left (\frac {\left (\textit {\_a}^{2}+2 c_{1} \right ) {\mathrm e}^{-1}}{2}\right )+1}d \textit {\_a} \right )-t -c_{2} = 0 \]
✓ Solution by Mathematica
Time used: 0.389 (sec). Leaf size: 126
DSolve[x''[t]-x[t]*Exp[x'[t]]==0,x[t],t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{-W\left (\frac {K[1]^2+2 c_1}{2 e}\right )-1}dK[1]\&\right ][t+c_2] \\ x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{-W\left (\frac {K[1]^2+2 (-1) c_1}{2 e}\right )-1}dK[1]\&\right ][t+c_2] \\ x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{-W\left (\frac {K[1]^2+2 c_1}{2 e}\right )-1}dK[1]\&\right ][t+c_2] \\ \end{align*}