Internal problem ID [6028]
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.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class A], _rational, _dAlembert]
Solve \begin {gather*} \boxed {\left (x^{2}+y^{2}\right )^{2} \left (y^{\prime }\right )^{2}-4 y^{2} x^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.203 (sec). Leaf size: 301
dsolve((x^2+y(x)^2)^2*diff(y(x),x)^2=4*x^2*y(x)^2,y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {-1+\sqrt {4 x^{2} c_{1}^{2}+1}}{2 c_{1}} \\ y \relax (x ) = \frac {1+\sqrt {4 x^{2} c_{1}^{2}+1}}{2 c_{1}} \\ y \relax (x ) = \frac {\frac {\left (4+4 \sqrt {4 x^{6} c_{1}^{3}+1}\right )^{\frac {1}{3}}}{2}-\frac {2 x^{2} c_{1}}{\left (4+4 \sqrt {4 x^{6} c_{1}^{3}+1}\right )^{\frac {1}{3}}}}{\sqrt {c_{1}}} \\ y \relax (x ) = \frac {-\frac {\left (4+4 \sqrt {4 x^{6} c_{1}^{3}+1}\right )^{\frac {1}{3}}}{4}+\frac {x^{2} c_{1}}{\left (4+4 \sqrt {4 x^{6} c_{1}^{3}+1}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (4+4 \sqrt {4 x^{6} c_{1}^{3}+1}\right )^{\frac {1}{3}}}{2}+\frac {2 x^{2} c_{1}}{\left (4+4 \sqrt {4 x^{6} c_{1}^{3}+1}\right )^{\frac {1}{3}}}\right )}{2}}{\sqrt {c_{1}}} \\ y \relax (x ) = \frac {-\frac {\left (4+4 \sqrt {4 x^{6} c_{1}^{3}+1}\right )^{\frac {1}{3}}}{4}+\frac {x^{2} c_{1}}{\left (4+4 \sqrt {4 x^{6} c_{1}^{3}+1}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (4+4 \sqrt {4 x^{6} c_{1}^{3}+1}\right )^{\frac {1}{3}}}{2}+\frac {2 x^{2} c_{1}}{\left (4+4 \sqrt {4 x^{6} c_{1}^{3}+1}\right )^{\frac {1}{3}}}\right )}{2}}{\sqrt {c_{1}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 20.97 (sec). Leaf size: 306
DSolve[(x^2+y[x]^2)^2*(y'[x])^2==4*x^2*y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} \left (-\sqrt {4 x^2+e^{2 c_1}}-e^{c_1}\right ) \\ y(x)\to \frac {1}{2} \left (\sqrt {4 x^2+e^{2 c_1}}-e^{c_1}\right ) \\ y(x)\to \frac {\sqrt [3]{\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}}}{\sqrt [3]{2}}-\frac {\sqrt [3]{2} x^2}{\sqrt [3]{\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}}} \\ y(x)\to \frac {2 \sqrt [3]{-2} x^2+(-2)^{2/3} \left (\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}\right ){}^{2/3}}{2 \sqrt [3]{\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}}} \\ y(x)\to -\frac {2 (-1)^{2/3} x^2+\sqrt [3]{-2} \left (\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}\right ){}^{2/3}}{2^{2/3} \sqrt [3]{\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}}} \\ y(x)\to 0 \\ \end{align*}