Internal problem ID [5047]
Book: Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold
Scientific. Singapore. 1995
Section: Chapter 1. First order differential equations. Section 1.2 Homogeneous equations problems.
page 12
Problem number: 48.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class G]]
Solve \begin {gather*} \boxed {\frac {2 x y y^{\prime }}{3}-\sqrt {x^{6}-y^{4}}-y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 100
dsolve(2/3*x*y(x)*diff(y(x),x)=sqrt(x^6-y(x)^4)+y(x)^2,y(x), singsol=all)
\[ \int _{\textit {\_b}}^{x}-\frac {\sqrt {\textit {\_a}^{6}-y \relax (x )^{4}}+y \relax (x )^{2}}{\sqrt {\textit {\_a}^{6}-y \relax (x )^{4}}\, \textit {\_a}}d \textit {\_a} +\int _{}^{y \relax (x )}\frac {2 \textit {\_f} \left (3 \sqrt {x^{6}-\textit {\_f}^{4}}\, \left (\int _{\textit {\_b}}^{x}\frac {\textit {\_a}^{5}}{\left (\textit {\_a}^{6}-\textit {\_f}^{4}\right )^{\frac {3}{2}}}d \textit {\_a} \right )+1\right )}{3 \sqrt {x^{6}-\textit {\_f}^{4}}}d \textit {\_f} +c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 9.032 (sec). Leaf size: 128
DSolve[2/3*x*y[x]*y'[x]==Sqrt[x^6-y[x]^4]+y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {x^{3/2}}{\sqrt [4]{\sec ^2\left (-\frac {\log \left (x^6\right )}{2}+3 c_1\right )}} \\ y(x)\to -\frac {i x^{3/2}}{\sqrt [4]{\sec ^2\left (-\frac {\log \left (x^6\right )}{2}+3 c_1\right )}} \\ y(x)\to \frac {i x^{3/2}}{\sqrt [4]{\sec ^2\left (-\frac {\log \left (x^6\right )}{2}+3 c_1\right )}} \\ y(x)\to \frac {x^{3/2}}{\sqrt [4]{\sec ^2\left (-\frac {\log \left (x^6\right )}{2}+3 c_1\right )}} \\ \end{align*}