23.2.325 problem 353

Internal problem ID [5680]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 353
Date solved : Tuesday, September 30, 2025 at 01:57:00 PM
CAS classification : [_quadrature]

\begin{align*} 3 {y^{\prime }}^{5}-y y^{\prime }+1&=0 \end{align*}

✓ Maple. Time used: 0.032 (sec). Leaf size: 87

ode:=3*diff(y(x),x)^5-y(x)*diff(y(x),x)+1 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {5 \operatorname {RootOf}\left (1+8 \textit {\_Z}^{5}+\left (2 c_1 -2 x \right ) \textit {\_Z}^{2}\right )^{3}+2 c_1 -2 x}{2 \operatorname {RootOf}\left (1+8 \textit {\_Z}^{5}+\left (2 c_1 -2 x \right ) \textit {\_Z}^{2}\right ) \left (4 \operatorname {RootOf}\left (1+8 \textit {\_Z}^{5}+\left (2 c_1 -2 x \right ) \textit {\_Z}^{2}\right )^{3}+c_1 -x \right )} \]

✓ Mathematica. Time used: 0.074 (sec). Leaf size: 176

ode=3*(D[y[x],x])^5 -y[x]*D[y[x],x]+1==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solve}\left [\int _1^{y(x)}\frac {1}{\text {Root}\left [3 \text {$\#$1}^5-K[1] \text {$\#$1}+1\&,1\right ]}dK[1]=x+c_1,y(x)\right ]\\ \text {Solve}\left [\int _1^{y(x)}\frac {1}{\text {Root}\left [3 \text {$\#$1}^5-K[2] \text {$\#$1}+1\&,2\right ]}dK[2]=x+c_1,y(x)\right ]\\ \text {Solve}\left [\int _1^{y(x)}\frac {1}{\text {Root}\left [3 \text {$\#$1}^5-K[3] \text {$\#$1}+1\&,3\right ]}dK[3]=x+c_1,y(x)\right ]\\ \text {Solve}\left [\int _1^{y(x)}\frac {1}{\text {Root}\left [3 \text {$\#$1}^5-K[4] \text {$\#$1}+1\&,4\right ]}dK[4]=x+c_1,y(x)\right ]\\ \text {Solve}\left [\int _1^{y(x)}\frac {1}{\text {Root}\left [3 \text {$\#$1}^5-K[5] \text {$\#$1}+1\&,5\right ]}dK[5]=x+c_1,y(x)\right ] \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-y(x)*Derivative(y(x), x) + 3*Derivative(y(x), x)**5 + 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -y(x)*Derivative(y(x), x) + 3*Derivative(y(x), x)**5 + 1 cannot be solved by the lie group method