23.2.247 problem 253

Internal problem ID [5602]
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 : 253
Date solved : Tuesday, September 30, 2025 at 01:11:17 PM
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: 0.287 (sec). Leaf size: 171

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} \\ \left (\ln \left (\frac {y^{2} \left (\operatorname {csgn}\left (y^{2}\right ) \sqrt {y^{4}-4 a^{2} x^{2}}+y^{2}\right )}{x}\right )+\ln \left (2\right )-2 \ln \left (y\right )\right ) \operatorname {csgn}\left (y^{2}\right )+c_1 -\ln \left (x \right ) &= 0 \\ \left (-\ln \left (\frac {y^{2} \left (\operatorname {csgn}\left (y^{2}\right ) \sqrt {y^{4}-4 a^{2} x^{2}}+y^{2}\right )}{x}\right )-\ln \left (2\right )+2 \ln \left (y\right )\right ) \operatorname {csgn}\left (y^{2}\right )+c_1 -\ln \left (x \right ) &= 0 \\ \end{align*}

✓ Mathematica. Time used: 13.183 (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