54.8.8 problem 1844

Internal problem ID [13068]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 7, non-linear third and higher order
Problem number : 1844
Date solved : Friday, October 03, 2025 at 03:59:11 AM
CAS classification : [[_3rd_order, _missing_x], [_3rd_order, _with_linear_symmetries]]

\begin{align*} 4 y^{2} y^{\prime \prime \prime }-18 y y^{\prime } y^{\prime \prime }+15 {y^{\prime }}^{3}&=0 \end{align*}

✓ Maple. Time used: 0.111 (sec). Leaf size: 77

ode:=4*y(x)^2*diff(diff(diff(y(x),x),x),x)-18*y(x)*diff(y(x),x)*diff(diff(y(x),x),x)+15*diff(y(x),x)^3 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ y &= {\mathrm e}^{\int \operatorname {RootOf}\left (-2 \int _{}^{\textit {\_Z}}\frac {1}{\textit {\_h}^{2}+\sqrt {c_1 \left (\textit {\_h}^{2}+c_1 \right )}+c_1}d \textit {\_h} +x +c_2 \right )d x +c_3} \\ y &= {\mathrm e}^{\int \operatorname {RootOf}\left (2 \int _{}^{\textit {\_Z}}-\frac {1}{\textit {\_h}^{2}-\sqrt {c_1 \left (\textit {\_h}^{2}+c_1 \right )}+c_1}d \textit {\_h} +x +c_2 \right )d x +c_3} \\ \end{align*}

✓ Mathematica. Time used: 0.068 (sec). Leaf size: 19

ode=15*D[y[x],x]^3 - 18*y[x]*D[y[x],x]*D[y[x],{x,2}] + 4*y[x]^2*Derivative[3][y][x] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{(x (c_3 x+c_2)+c_1){}^2} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*y(x)**2*Derivative(y(x), (x, 3)) - 18*y(x)*Derivative(y(x), x)*Derivative(y(x), (x, 2)) + 15*Derivative(y(x), x)**3,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE (sqrt(1296*y(x)**4*Derivative(y(x), (x, 3))**2/25 - 23328*y(x)**3*Derivative(y(x), (x, 2))**3/125)/2 + 18*y(x)**2*Derivative(y(x), (x, 3))/5)**(1/3)/3 + Derivative(y(x), x) + 6*y(x)*Derivative(y(x), (x, 2))/(5*(sqrt(1296*y(x)**4*Derivative(y(x), (x, 3))**2/25 - 23328*y(x)**3*Derivative(y(x), (x, 2))**3/125)/2 + 18*y(x)**2*Derivative(y(x), (x, 3))/5)**(1/3)) cannot be solved by the factorable group method