75.21.9 problem 704

Internal problem ID [17107]
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 : 704
Date solved : Monday, March 31, 2025 at 03:41:55 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} x^{\prime \prime }+\left (x+2\right ) x^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.050 (sec). Leaf size: 30

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

\[ x = \frac {\left (-\sqrt {2}\, c_1 +\tanh \left (\frac {\left (t +c_2 \right ) \sqrt {2}}{2 c_1}\right )\right ) \sqrt {2}}{c_1} \]

✓ Mathematica. Time used: 0.138 (sec). Leaf size: 108

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

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

✗ Sympy

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

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