58.1.46 problem 46

Internal problem ID [9117]
Book : Second order enumerated odes
Section : section 1
Problem number : 46
Date solved : Sunday, March 30, 2025 at 02:07:39 PM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 1.138 (sec). Leaf size: 178

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

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

✓ Mathematica. Time used: 1.974 (sec). Leaf size: 361

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

\begin{align*} y(x)\to \text {InverseFunction}\left [-4 \left (\frac {1}{2} \log \left (2 \sqrt {\text {$\#$1}}-i c_1\right )-\frac {i c_1}{2 \left (2 \sqrt {\text {$\#$1}}-i c_1\right )}\right )\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [-4 \left (\frac {i c_1}{2 \left (2 \sqrt {\text {$\#$1}}+i c_1\right )}+\frac {1}{2} \log \left (2 \sqrt {\text {$\#$1}}+i c_1\right )\right )\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [-4 \left (\frac {1}{2} \log \left (2 \sqrt {\text {$\#$1}}-i (-c_1)\right )-\frac {i (-c_1)}{2 \left (2 \sqrt {\text {$\#$1}}-i (-c_1)\right )}\right )\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [-4 \left (\frac {i (-c_1)}{2 \left (2 \sqrt {\text {$\#$1}}+i (-1) c_1\right )}+\frac {1}{2} \log \left (2 \sqrt {\text {$\#$1}}+i (-1) c_1\right )\right )\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [-4 \left (\frac {1}{2} \log \left (2 \sqrt {\text {$\#$1}}-i c_1\right )-\frac {i c_1}{2 \left (2 \sqrt {\text {$\#$1}}-i c_1\right )}\right )\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [-4 \left (\frac {i c_1}{2 \left (2 \sqrt {\text {$\#$1}}+i c_1\right )}+\frac {1}{2} \log \left (2 \sqrt {\text {$\#$1}}+i c_1\right )\right )\&\right ][x+c_2] \\ \end{align*}

✗ Sympy

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

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