60.7.136 problem 1751 (book 6.160)

Internal problem ID [11686]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1751 (book 6.160)
Date solved : Sunday, March 30, 2025 at 08:42:24 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 4 y^{\prime \prime } y-3 {y^{\prime }}^{2}+a y^{3}+y^{2} b +c y&=0 \end{align*}

✓ Maple. Time used: 0.043 (sec). Leaf size: 90

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

\begin{align*} y &= 0 \\ -\sqrt {3}\, \int _{}^{y}\frac {1}{\sqrt {\textit {\_a} \left (3 \sqrt {\textit {\_a}}\, c_1 -\textit {\_a}^{2} a -3 \textit {\_a} b +3 c \right )}}d \textit {\_a} -x -c_2 &= 0 \\ \sqrt {3}\, \int _{}^{y}\frac {1}{\sqrt {\textit {\_a} \left (3 \sqrt {\textit {\_a}}\, c_1 -\textit {\_a}^{2} a -3 \textit {\_a} b +3 c \right )}}d \textit {\_a} -x -c_2 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 1.441 (sec). Leaf size: 323

ode=c*y[x] + b*y[x]^2 + a*y[x]^3 - 3*D[y[x],x]^2 + 4*y[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 {$\#$1}}-\frac {1}{\sqrt {-\frac {1}{3} a K[1]^3-b K[1]^2+c_1 K[1]^{3/2}+c K[1]}}dK[1]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\sqrt {-\frac {1}{3} a K[2]^3-b K[2]^2+c_1 K[2]^{3/2}+c K[2]}}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {1}{\sqrt {-\frac {1}{3} a K[1]^3-b K[1]^2-c_1 K[1]^{3/2}+c K[1]}}dK[1]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {1}{\sqrt {-\frac {1}{3} a K[1]^3-b K[1]^2+c_1 K[1]^{3/2}+c K[1]}}dK[1]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\sqrt {-\frac {1}{3} a K[2]^3-b K[2]^2-c_1 K[2]^{3/2}+c K[2]}}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\sqrt {-\frac {1}{3} a K[2]^3-b K[2]^2+c_1 K[2]^{3/2}+c K[2]}}dK[2]\&\right ][x+c_2] \\ \end{align*}

✗ Sympy

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

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