54.7.170 problem 1788 (book 6.197)

Internal problem ID [13019]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1788 (book 6.197)
Date solved : Wednesday, October 01, 2025 at 02:57:11 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 2 y \left (1-y\right ) y^{\prime \prime }-\left (1-3 y\right ) {y^{\prime }}^{2}+h \left (y\right )&=0 \end{align*}

✓ Maple. Time used: 0.016 (sec). Leaf size: 81

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

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

✓ Mathematica. Time used: 0.495 (sec). Leaf size: 710

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
h = Function("h") 
ode = Eq((2 - 2*y(x))*y(x)*Derivative(y(x), (x, 2)) + (3*y(x) - 1)*Derivative(y(x), x)**2 + h(y(x)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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