Internal problem ID [15443]
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: 699.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
\[ \boxed {x^{\prime \prime }+{x^{\prime }}^{2}+x=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 61
dsolve(diff(x(t),t$2)+diff(x(t),t)^2+x(t)=0,x(t), singsol=all)
\begin{align*} -2 \left (\int _{}^{x \left (t \right )}\frac {1}{\sqrt {2+4 \,{\mathrm e}^{-2 \textit {\_a}} c_{1} -4 \textit {\_a}}}d \textit {\_a} \right )-t -c_{2} &= 0 \\ 2 \left (\int _{}^{x \left (t \right )}\frac {1}{\sqrt {2+4 \,{\mathrm e}^{-2 \textit {\_a}} c_{1} -4 \textit {\_a}}}d \textit {\_a} \right )-t -c_{2} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.81 (sec). Leaf size: 272
DSolve[x''[t]+x'[t]^2+x[t]==0,x[t],t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\sqrt {2}}{\sqrt {2 e^{-2 K[1]} c_1-2 K[1]+1}}dK[1]\&\right ][t+c_2] \\ x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt {2}}{\sqrt {2 e^{-2 K[2]} c_1-2 K[2]+1}}dK[2]\&\right ][t+c_2] \\ x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\sqrt {2}}{\sqrt {2 e^{-2 K[1]} (-c_1)-2 K[1]+1}}dK[1]\&\right ][t+c_2] \\ x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\sqrt {2}}{\sqrt {2 e^{-2 K[1]} c_1-2 K[1]+1}}dK[1]\&\right ][t+c_2] \\ x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt {2}}{\sqrt {2 e^{-2 K[2]} (-c_1)-2 K[2]+1}}dK[2]\&\right ][t+c_2] \\ x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt {2}}{\sqrt {2 e^{-2 K[2]} c_1-2 K[2]+1}}dK[2]\&\right ][t+c_2] \\ \end{align*}