75.21.8 problem 703

Internal problem ID [17177]
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 : 703
Date solved : Tuesday, January 28, 2025 at 09:56:13 AM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} x^{\prime \prime }+x {x^{\prime }}^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.026 (sec). Leaf size: 31

dsolve(diff(x(t),t$2)+x(t)*diff(x(t),t)^2=0,x(t), singsol=all)
 

\[ x \left (t \right ) = -i \operatorname {RootOf}\left (i \sqrt {2}\, c_{1} t +i \sqrt {2}\, c_{2} -\operatorname {erf}\left (\textit {\_Z} \right ) \sqrt {\pi }\right ) \sqrt {2} \]

✓ Solution by Mathematica

Time used: 1.673 (sec). Leaf size: 34

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

\[ x(t)\to -i \sqrt {2} \text {erf}^{-1}\left (i \sqrt {\frac {2}{\pi }} c_1 (t+c_2)\right ) \]