75.21.6 problem 701

Internal problem ID [17175]
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 ODEs). Section 16. The method of isoclines for differential equations of the second order. Exercises page 158
Problem number : 701
Date solved : Tuesday, January 28, 2025 at 09:56:10 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} x^{\prime \prime }-x \,{\mathrm e}^{x^{\prime }}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.048 (sec). Leaf size: 33

dsolve(diff(x(t),t$2)-x(t)*exp(diff(x(t),t))=0,x(t), singsol=all)
 

\[ -\int _{}^{x \left (t \right )}\frac {1}{\operatorname {LambertW}\left (\frac {\left (\textit {\_a}^{2}+2 c_{1} \right ) {\mathrm e}^{-1}}{2}\right )+1}d \textit {\_a} -t -c_{2} = 0 \]

✓ Solution by Mathematica

Time used: 0.171 (sec). Leaf size: 48

DSolve[D[x[t],{t,2}]-x[t]*Exp[D[x[t],t]]==0,x[t],t,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{x(t)}\frac {1}{\text {InverseFunction}\left [\int _1^{\text {$\#$1}}e^{-K[1]} K[1]dK[1]\&\right ]\left [\frac {K[2]^2}{2}+c_1\right ]}dK[2]=t+c_2,x(t)\right ] \]