29.34.8 problem 1004

Internal problem ID [5579]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 34
Problem number : 1004
Date solved : Sunday, March 30, 2025 at 09:04:23 AM
CAS classification : [[_homogeneous, `class A`], _rational, _Bernoulli]

\begin{align*} 4 x^{2} y^{2} {y^{\prime }}^{2}&=\left (x^{2}+y^{2}\right )^{2} \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 69

ode:=4*x^2*y(x)^2*diff(y(x),x)^2 = (x^2+y(x)^2)^2; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \sqrt {\left (x +c_1 \right ) x} \\ y &= -\sqrt {\left (x +c_1 \right ) x} \\ y &= -\frac {\sqrt {3}\, \sqrt {-x \left (x^{3}-3 c_1 \right )}}{3 x} \\ y &= \frac {\sqrt {3}\, \sqrt {-x \left (x^{3}-3 c_1 \right )}}{3 x} \\ \end{align*}

✓ Mathematica. Time used: 0.541 (sec). Leaf size: 97

ode=4 x^2 y[x]^2(D[y[x],x])^2 ==(x^2+y[x]^2)^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\sqrt {x} \sqrt {x+c_1} \\ y(x)\to \sqrt {x} \sqrt {x+c_1} \\ y(x)\to -\frac {\sqrt {-x^3+3 c_1}}{\sqrt {3} \sqrt {x}} \\ y(x)\to \frac {\sqrt {-x^3+3 c_1}}{\sqrt {3} \sqrt {x}} \\ \end{align*}

✓ Sympy. Time used: 1.456 (sec). Leaf size: 61

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x**2*y(x)**2*Derivative(y(x), x)**2 - (x**2 + y(x)**2)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = - \sqrt {x \left (C_{1} + x\right )}, \ y{\left (x \right )} = \sqrt {x \left (C_{1} + x\right )}, \ y{\left (x \right )} = - \frac {\sqrt {3} \sqrt {\frac {C_{1}}{x} - x^{2}}}{3}, \ y{\left (x \right )} = \frac {\sqrt {3} \sqrt {\frac {C_{1}}{x} - x^{2}}}{3}\right ] \]