Internal problem ID [6781]
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]
\[ \boxed {\left (x^{2}+y^{2}\right )^{2} {y^{\prime }}^{2}-4 y^{2} x^{2}=0} \]
✓ Solution by Maple
Time used: 0.141 (sec). Leaf size: 255
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 \left (x \right ) &= \frac {1-\sqrt {4 x^{2} c_{1}^{2}+1}}{2 c_{1}} \\ y \left (x \right ) &= \frac {1+\sqrt {4 x^{2} c_{1}^{2}+1}}{2 c_{1}} \\ y \left (x \right ) &= -\frac {2 \left (c_{1} x^{2}-\frac {\left (4+4 \sqrt {4 c_{1}^{3} x^{6}+1}\right )^{\frac {2}{3}}}{4}\right )}{\sqrt {c_{1}}\, \left (4+4 \sqrt {4 c_{1}^{3} x^{6}+1}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (4+4 \sqrt {4 c_{1}^{3} x^{6}+1}\right )^{\frac {1}{3}}}{4 \sqrt {c_{1}}}-\frac {\left (i \sqrt {3}-1\right ) x^{2} \sqrt {c_{1}}}{\left (4+4 \sqrt {4 c_{1}^{3} x^{6}+1}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {4 i \sqrt {3}\, c_{1} x^{2}+i \sqrt {3}\, \left (4+4 \sqrt {4 c_{1}^{3} x^{6}+1}\right )^{\frac {2}{3}}+4 c_{1} x^{2}-\left (4+4 \sqrt {4 c_{1}^{3} x^{6}+1}\right )^{\frac {2}{3}}}{4 \left (4+4 \sqrt {4 c_{1}^{3} x^{6}+1}\right )^{\frac {1}{3}} \sqrt {c_{1}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 15.845 (sec). Leaf size: 345
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 {i 2^{2/3} \left (\sqrt {3}+i\right ) \left (\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}\right ){}^{2/3}+\sqrt [3]{2} \left (2+2 i \sqrt {3}\right ) x^2}{4 \sqrt [3]{\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}}} \\ y(x)\to \frac {\left (1-i \sqrt {3}\right ) x^2}{2^{2/3} \sqrt [3]{\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}}}{2 \sqrt [3]{2}} \\ y(x)\to 0 \\ \end{align*}