47.1.15 problem 15

Internal problem ID [9735]
Book : Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam Publishing Co. NY. 6th edition. 1981.
Section : CHAPTER 16. Nonlinear equations. Section 94. Factoring the left member. EXERCISES Page 309
Problem number : 15
Date solved : Tuesday, September 30, 2025 at 06:32:29 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.139 (sec). Leaf size: 251

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

\begin{align*} y &= \frac {1-\sqrt {4 c_1^{2} x^{2}+1}}{2 c_1} \\ y &= \frac {1+\sqrt {4 c_1^{2} x^{2}+1}}{2 c_1} \\ y &= -\frac {2 \left (c_1 \,x^{2}-\frac {\left (4+4 \sqrt {4 c_1^{3} x^{6}+1}\right )^{{2}/{3}}}{4}\right )}{\left (4+4 \sqrt {4 c_1^{3} x^{6}+1}\right )^{{1}/{3}} \sqrt {c_1}} \\ y &= -\frac {\left (1+i \sqrt {3}\right ) \left (4+4 \sqrt {4 c_1^{3} x^{6}+1}\right )^{{1}/{3}}}{4 \sqrt {c_1}}-\frac {x^{2} \sqrt {c_1}\, \left (i \sqrt {3}-1\right )}{\left (4+4 \sqrt {4 c_1^{3} x^{6}+1}\right )^{{1}/{3}}} \\ y &= \frac {4 i \sqrt {3}\, c_1 \,x^{2}+i \left (4+4 \sqrt {4 c_1^{3} x^{6}+1}\right )^{{2}/{3}} \sqrt {3}+4 c_1 \,x^{2}-\left (4+4 \sqrt {4 c_1^{3} x^{6}+1}\right )^{{2}/{3}}}{4 \left (4+4 \sqrt {4 c_1^{3} x^{6}+1}\right )^{{1}/{3}} \sqrt {c_1}} \\ \end{align*}

✓ Mathematica. Time used: 0.315 (sec). Leaf size: 94

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

\begin{align*} \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {K[1]^2+1}{(K[1]-1) K[1] (K[1]+1)}dK[1]=-\log (x)+c_1,y(x)\right ]\\ \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {K[2]^2+1}{K[2] \left (K[2]^2+3\right )}dK[2]=-\log (x)+c_1,y(x)\right ]\\ y(x)&\to 0 \end{align*}

✗ Sympy

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

Timed Out