29.34.5 problem 1000

Internal problem ID [5576]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 34
Problem number : 1000
Date solved : Sunday, March 30, 2025 at 09:03:56 AM
CAS classification : [[_homogeneous, `class G`], _rational]

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

✓ Maple. Time used: 0.395 (sec). Leaf size: 141

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

\begin{align*} y &= \sqrt {2}\, \sqrt {-x a} \\ y &= -\sqrt {2}\, \sqrt {-x a} \\ y &= \sqrt {2}\, \sqrt {x a} \\ y &= -\sqrt {2}\, \sqrt {x a} \\ y &= \frac {{\mathrm e}^{\frac {c_1}{2}+\frac {\operatorname {RootOf}\left (\left (16 x \,a^{2} {\mathrm e}^{2 c_1}+x^{3}-4 \,{\mathrm e}^{\textit {\_Z} +2 c_1}\right ) {\mathrm e}^{2 \textit {\_Z}}\right )}{2}}}{\sqrt {x}} \\ y &= \sqrt {x}\, {\mathrm e}^{-\frac {c_1}{2}+\frac {\operatorname {RootOf}\left (x^{2} \left (16 a^{2} x^{2} {\mathrm e}^{-2 c_1}+1-4 x \,{\mathrm e}^{\textit {\_Z} -2 c_1}\right ) {\mathrm e}^{2 \textit {\_Z}}\right )}{2}} \\ \end{align*}

✓ Mathematica. Time used: 18.764 (sec). Leaf size: 219

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

\begin{align*} y(x)\to -\sqrt {-2 a^2 e^{-c_1} x^2-\frac {e^{c_1}}{2}} \\ y(x)\to \sqrt {-2 a^2 e^{-c_1} x^2-\frac {e^{c_1}}{2}} \\ y(x)\to -\frac {\sqrt {4 a^2 e^{-c_1} x^2+e^{c_1}}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {4 a^2 e^{-c_1} x^2+e^{c_1}}}{\sqrt {2}} \\ y(x)\to -\sqrt {2} \sqrt {a} \sqrt {x} \\ y(x)\to -i \sqrt {2} \sqrt {a} \sqrt {x} \\ y(x)\to i \sqrt {2} \sqrt {a} \sqrt {x} \\ y(x)\to \sqrt {2} \sqrt {a} \sqrt {x} \\ \end{align*}

✗ Sympy

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

Timed Out