29.29.2 problem 824

Internal problem ID [5407]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 29
Problem number : 824
Date solved : Tuesday, March 04, 2025 at 09:35:54 PM
CAS classification : [[_homogeneous, `class G`]]

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

✓ Maple. Time used: 0.294 (sec). Leaf size: 124

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

\begin{align*} y \left (x \right ) &= \frac {4}{x^{2}} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {2 \sqrt {2}\, x -2 c_{1}}{c_{1} \left (c_{1}^{2}-2 x^{2}\right )} \\ y \left (x \right ) &= \frac {-2 \sqrt {2}\, x -2 c_{1}}{c_{1} \left (c_{1}^{2}-2 x^{2}\right )} \\ y \left (x \right ) &= -\frac {\left (c_{1} \sqrt {2}\, x -2\right ) c_{1}^{2}}{2 c_{1}^{2} x^{2}-4} \\ y \left (x \right ) &= \frac {\left (c_{1} \sqrt {2}\, x +2\right ) c_{1}^{2}}{2 c_{1}^{2} x^{2}-4} \\ \end{align*}

✓ Mathematica. Time used: 0.959 (sec). Leaf size: 152

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

\begin{align*} \text {Solve}\left [-\frac {\sqrt {y(x)} \sqrt {4-x^2 y(x)} \text {arcsinh}\left (\frac {1}{2} x \sqrt {-y(x)}\right )}{\sqrt {-y(x)} \sqrt {x^2 y(x)-4}}-\frac {1}{2} \log (y(x))&=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {\sqrt {y(x)} \sqrt {4-x^2 y(x)} \text {arcsinh}\left (\frac {1}{2} x \sqrt {-y(x)}\right )}{\sqrt {-y(x)} \sqrt {x^2 y(x)-4}}-\frac {1}{2} \log (y(x))&=c_1,y(x)\right ] \\ y(x)\to 0 \\ y(x)\to \frac {4}{x^2} \\ \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE x*y(x)**2/2 - sqrt((x**2*y(x) - 4)*y(x)**3)/2 + Derivative(y(x), x) cannot be solved by the factorable group method