75.21.4 problem 699

Internal problem ID [17102]
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 : 699
Date solved : Monday, March 31, 2025 at 03:41:42 PM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 0.022 (sec). Leaf size: 61

ode:=diff(diff(x(t),t),t)+diff(x(t),t)^2+x(t) = 0; 
dsolve(ode,x(t), singsol=all);
 

\begin{align*} -2 \int _{}^{x}\frac {1}{\sqrt {2+4 \,{\mathrm e}^{-2 \textit {\_a}} c_1 -4 \textit {\_a}}}d \textit {\_a} -t -c_2 &= 0 \\ 2 \int _{}^{x}\frac {1}{\sqrt {2+4 \,{\mathrm e}^{-2 \textit {\_a}} c_1 -4 \textit {\_a}}}d \textit {\_a} -t -c_2 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.411 (sec). Leaf size: 308

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

\begin{align*} x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {e^{K[2]}}{\sqrt {c_1+2 \int _1^{K[2]}-e^{2 K[1]} K[1]dK[1]}}dK[2]\&\right ][t+c_2] \\ x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {e^{K[3]}}{\sqrt {c_1+2 \int _1^{K[3]}-e^{2 K[1]} K[1]dK[1]}}dK[3]\&\right ][t+c_2] \\ x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {e^{K[2]}}{\sqrt {2 \int _1^{K[2]}-e^{2 K[1]} K[1]dK[1]-c_1}}dK[2]\&\right ][t+c_2] \\ x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {e^{K[2]}}{\sqrt {c_1+2 \int _1^{K[2]}-e^{2 K[1]} K[1]dK[1]}}dK[2]\&\right ][t+c_2] \\ x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {e^{K[3]}}{\sqrt {2 \int _1^{K[3]}-e^{2 K[1]} K[1]dK[1]-c_1}}dK[3]\&\right ][t+c_2] \\ x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {e^{K[3]}}{\sqrt {c_1+2 \int _1^{K[3]}-e^{2 K[1]} K[1]dK[1]}}dK[3]\&\right ][t+c_2] \\ \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(x(t) + Derivative(x(t), t)**2 + Derivative(x(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

NotImplementedError : The given ODE -sqrt(-x(t) - Derivative(x(t), (t, 2))) + Derivative(x(t), t) cannot be solved by the factorable group method