Internal problem ID [6030]
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: 17.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class A], _rational, _Bernoulli]
Solve \begin {gather*} \boxed {x y \left (x^{2}+y^{2}\right ) \left (\left (y^{\prime }\right )^{2}-1\right )-y^{\prime } \left (x^{4}+y^{2} x^{2}+y^{4}\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.844 (sec). Leaf size: 250
dsolve(x*y(x)*(x^2+y(x)^2)*(diff(y(x),x)^2-1)=diff(y(x),x)*(x^4+x^2*y(x)^2+y(x)^4),y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\sqrt {x^{2} c_{1} \left (c_{1} x^{2}+\sqrt {x^{4} c_{1}^{2}+1}\right )}}{x \left (c_{1} x^{2}+\sqrt {x^{4} c_{1}^{2}+1}\right ) c_{1}} \\ y \relax (x ) = \frac {\sqrt {-x^{2} c_{1} \left (-c_{1} x^{2}+\sqrt {x^{4} c_{1}^{2}+1}\right )}}{x \left (c_{1} x^{2}-\sqrt {x^{4} c_{1}^{2}+1}\right ) c_{1}} \\ y \relax (x ) = -\frac {\sqrt {x^{2} c_{1} \left (c_{1} x^{2}+\sqrt {x^{4} c_{1}^{2}+1}\right )}}{x \left (c_{1} x^{2}+\sqrt {x^{4} c_{1}^{2}+1}\right ) c_{1}} \\ y \relax (x ) = -\frac {\sqrt {-x^{2} c_{1} \left (-c_{1} x^{2}+\sqrt {x^{4} c_{1}^{2}+1}\right )}}{x \left (c_{1} x^{2}-\sqrt {x^{4} c_{1}^{2}+1}\right ) c_{1}} \\ y \relax (x ) = \sqrt {2 \ln \relax (x )+c_{1}}\, x \\ y \relax (x ) = -\sqrt {2 \ln \relax (x )+c_{1}}\, x \\ \end{align*}
✓ Solution by Mathematica
Time used: 10.153 (sec). Leaf size: 248
DSolve[x*y[x]*(x^2+y[x]^2)*((y'[x])^2-1)==y'[x]*(x^4+x^2*y[x]^2+y[x]^4),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {-x^2-\sqrt {x^4+e^{4 c_1}}} \\ y(x)\to \sqrt {-x^2-\sqrt {x^4+e^{4 c_1}}} \\ y(x)\to -\sqrt {-x^2+\sqrt {x^4+e^{4 c_1}}} \\ y(x)\to \sqrt {-x^2+\sqrt {x^4+e^{4 c_1}}} \\ y(x)\to -x \sqrt {2 \log (x)+c_1} \\ y(x)\to x \sqrt {2 \log (x)+c_1} \\ y(x)\to -\sqrt {-\sqrt {x^4}-x^2} \\ y(x)\to \sqrt {-\sqrt {x^4}-x^2} \\ y(x)\to -\sqrt {\sqrt {x^4}-x^2} \\ y(x)\to \sqrt {\sqrt {x^4}-x^2} \\ \end{align*}