71.1.100 problem 128 (page 179)

Internal problem ID [19276]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 128 (page 179)
Date solved : Friday, October 03, 2025 at 07:34:09 AM
CAS classification : [[_high_order, _missing_x], [_high_order, _missing_y], [_high_order, _with_linear_symmetries]]

\begin{align*} 40 {y^{\prime \prime \prime }}^{3}-45 y^{\prime \prime } y^{\prime \prime \prime } y^{\prime \prime \prime \prime }+9 {y^{\prime \prime }}^{2} y^{\left (5\right )}&=0 \end{align*}

✓ Maple. Time used: 0.156 (sec). Leaf size: 113

ode:=40*diff(diff(diff(y(x),x),x),x)^3-45*diff(diff(y(x),x),x)*diff(diff(diff(y(x),x),x),x)*diff(diff(diff(diff(y(x),x),x),x),x)+9*diff(diff(y(x),x),x)^2*diff(diff(diff(diff(diff(y(x),x),x),x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= c_1 x +c_2 \\ y &= \int \int \operatorname {RootOf}\left (-\int _{}^{\textit {\_Z}}\frac {1}{\operatorname {RootOf}\left (-\ln \left (\textit {\_f} \right )-6 \int _{}^{\textit {\_Z}}\frac {\textit {\_k}}{\sqrt {\textit {\_k}^{4}-c_1}}d \textit {\_k} +c_2 \right ) \textit {\_f}^{{3}/{2}}}d \textit {\_f} +x +c_3 \right )d x d x +c_4 x +c_5 \\ y &= \int \int \operatorname {RootOf}\left (-\int _{}^{\textit {\_Z}}\frac {1}{\operatorname {RootOf}\left (-\ln \left (\textit {\_f} \right )+6 \int _{}^{\textit {\_Z}}\frac {\textit {\_k}}{\sqrt {\textit {\_k}^{4}-c_1}}d \textit {\_k} +c_2 \right ) \textit {\_f}^{{3}/{2}}}d \textit {\_f} +x +c_3 \right )d x d x +c_4 x +c_5 \\ \end{align*}

✓ Mathematica. Time used: 0.161 (sec). Leaf size: 43

ode=40*D[y[x],{x,3}]^3-45*D[y[x],{x,2}]*D[y[x],{x,3}]*D[y[x],{x,4}]+9*D[y[x],{x,2}]^2*D[y[x],{x,5}]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_5 x-\frac {4 \sqrt {x (c_3 x+c_2)+c_1}}{c_2{}^2-4 c_1 c_3}+c_4 \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(9*Derivative(y(x), (x, 2))**2*Derivative(y(x), (x, 5)) - 45*Derivative(y(x), (x, 2))*Derivative(y(x), (x, 3))*Derivative(y(x), (x, 4)) + 40*Derivative(y(x), (x, 3))**3,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE (sqrt(59049*Dummy_106(x)**4*Derivative(Dummy_106(x), (x, 3))**2/1600 - 19683*Dummy_106(x)**3*Derivative(Dummy_106(x), (x, 2))**3/128)/2 + 243*Dummy_106(x)**2*Derivative(Dummy_106(x), (x, 3))/80)**(1/3)/3 + Derivative(Dummy_106(x), x) + 9*Dummy_106(x)*Derivative(Dummy_106(x), (x, 2))/(8*(sqrt(59049*Dummy_106(x)**4*Derivative(Dummy_106(x), (x, 3))**2/1600 - 19683*Dummy_106(x)**3*Derivative(Dummy_106(x), (x, 2))**3/128)/2 + 243*Dummy_106(x)**2*Derivative(Dummy_106(x), (x, 3))/80)**(1/3)) cannot be solved by the factorable group method