60.7.108 problem 1721 (book 6.130)

Internal problem ID [11658]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1721 (book 6.130)
Date solved : Sunday, March 30, 2025 at 08:36:35 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime } y+a {y^{\prime }}^{2}+b y^{2} y^{\prime }+c y^{4}&=0 \end{align*}

✓ Maple. Time used: 0.035 (sec). Leaf size: 179

ode:=diff(diff(y(x),x),x)*y(x)+a*diff(y(x),x)^2+b*y(x)^2*diff(y(x),x)+c*y(x)^4 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ \left (2 a +4\right ) \int _{}^{y}\frac {1}{\tan \left (\operatorname {RootOf}\left (2 \textit {\_Z} b \,\textit {\_a}^{2}-2 a \ln \left (\textit {\_a} \right ) \sqrt {\textit {\_a}^{4} \left (4 a c -b^{2}+8 c \right )}-\sqrt {\textit {\_a}^{4} \left (4 a c -b^{2}+8 c \right )}\, \ln \left (\frac {\textit {\_a}^{4} \left (4 a c -b^{2}+8 c \right ) \sec \left (\textit {\_Z} \right )^{2}}{a +2}\right )+2 \sqrt {\textit {\_a}^{4} \left (4 a c -b^{2}+8 c \right )}\, \ln \left (2\right )+c_1 \sqrt {\textit {\_a}^{4} \left (4 a c -b^{2}+8 c \right )}\right )\right ) \sqrt {\textit {\_a}^{4} \left (4 \left (a +2\right ) c -b^{2}\right )}-b \,\textit {\_a}^{2}}d \textit {\_a} -x -c_2 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 50.939 (sec). Leaf size: 105

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

\[ \text {Solve}\left [\int _1^{y(x)}\frac {1}{K[2]^2 \text {InverseFunction}\left [\frac {\log (c+\text {$\#$1} (b+(a+2) \text {$\#$1}))-\frac {2 b \arctan \left (\frac {b+2 (a+2) \text {$\#$1}}{\sqrt {4 (a+2) c-b^2}}\right )}{\sqrt {4 (a+2) c-b^2}}}{2 (a+2)}\&\right ][c_1-\log (K[2])]}dK[2]=x-c_2,y(x)\right ] \]

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (-b*y(x)**2 + sqrt((-4*a*c*y(x)**3 - 4*a*Derivative(y(x), (x, 2)) + b**2*y(x)**3)*y(x)))/(2*a) cannot be solved by the factorable group method