60.7.54 problem 1658 (book 6.67)

Internal problem ID [11604]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1658 (book 6.67)
Date solved : Wednesday, March 05, 2025 at 02:34:39 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+y^{3} y^{\prime }-y y^{\prime } \sqrt {y^{4}+4 y^{\prime }}&=0 \end{align*}

✓ Maple. Time used: 0.142 (sec). Leaf size: 131

ode:=diff(diff(y(x),x),x)+y(x)^3*diff(y(x),x)-y(x)*diff(y(x),x)*(y(x)^4+4*diff(y(x),x))^(1/2) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {2^{{2}/{3}} \left (\left (4 c_{1} +3 x \right )^{2}\right )^{{1}/{3}}}{4 c_{1} +3 x} \\ y &= -\frac {2^{{2}/{3}} \left (\left (4 c_{1} +3 x \right )^{2}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{8 c_{1} +6 x} \\ y &= \frac {2^{{2}/{3}} \left (\left (4 c_{1} +3 x \right )^{2}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{8 c_{1} +6 x} \\ y &= \tan \left (\frac {x +c_{2}}{c_{1}^{3}}\right ) \sqrt {\frac {1}{c_{1}^{2}}} \\ y &= \tanh \left (\frac {x +c_{2}}{c_{1}^{3}}\right ) \sqrt {\frac {1}{c_{1}^{2}}} \\ \end{align*}

✓ Mathematica. Time used: 1.839 (sec). Leaf size: 317

ode=y[x]^3*D[y[x],x] - y[x]*D[y[x],x]*Sqrt[y[x]^4 + 4*D[y[x],x]] + D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-sqrt(y(x)**4 + 4*Derivative(y(x), x))*y(x)*Derivative(y(x), x) + y(x)**3*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE (sqrt(-27*y(x)**3*Derivative(y(x), (x, 2))**3/2 + 729*Derivative(y(x), (x, 2))**4/(16*y(x)**4))/2 - 27*Derivative(y(x), (x, 2))**2/(8*y(x)**2))**(1/3)/3 + Derivative(y(x), x) + y(x)*Derivative(y(x), (x, 2))/(2*(sqrt(-27*y(x)**3*Derivative(y(x), (x, 2))**3/2 + 729*Derivative(y(x), (x, 2))**4/(16*y(x)**4))/2 - 27*Derivative(y(x), (x, 2))**2/(8*y(x)**2))**(1/3)) cannot be solved by the factorable group method