47.2.52 problem 48

Internal problem ID [7468]
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
Date solved : Monday, January 27, 2025 at 03:01:22 PM
CAS classification : [[_homogeneous, `class G`]]

\begin{align*} \frac {2 x y y^{\prime }}{3}&=\sqrt {x^{6}-y^{4}}+y^{2} \end{align*}

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 102

dsolve(2/3*x*y(x)*diff(y(x),x)=sqrt(x^6-y(x)^4)+y(x)^2,y(x), singsol=all)
 

\[ -\int _{\textit {\_b}}^{x}\frac {\sqrt {\textit {\_a}^{6}-y^{4}}+y^{2}}{\sqrt {\textit {\_a}^{6}-y^{4}}\, \textit {\_a}}d \textit {\_a} +\frac {2 \left (\int _{}^{y}\frac {\textit {\_f} \left (3 \left (\int _{\textit {\_b}}^{x}\frac {\textit {\_a}^{5}}{\left (\textit {\_a}^{6}-\textit {\_f}^{4}\right )^{{3}/{2}}}d \textit {\_a} \right ) \sqrt {x^{6}-\textit {\_f}^{4}}+1\right )}{\sqrt {x^{6}-\textit {\_f}^{4}}}d \textit {\_f} \right )}{3}+c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 3.174 (sec). Leaf size: 209

DSolve[2/3*x*y[x]*D[y[x],x]==Sqrt[x^6-y[x]^4]+y[x]^2,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \left (-\frac {1}{2}-\frac {i}{2}\right ) e^{-\frac {3 i c_1}{2}} x^{\frac {3}{2}-\frac {3 i}{2}} \sqrt {e^{6 i c_1}-x^{6 i}} \\ y(x)\to \left (\frac {1}{2}+\frac {i}{2}\right ) e^{-\frac {3 i c_1}{2}} x^{\frac {3}{2}-\frac {3 i}{2}} \sqrt {e^{6 i c_1}-x^{6 i}} \\ y(x)\to \left (-\frac {1}{2}-\frac {i}{2}\right ) x^{\frac {3}{2}-\frac {3 i}{2}} e^{2 i \text {Interval}[\{0,\pi \}]} \sqrt {e^{2 i \text {Interval}[\{0,\pi \}]}-x^{6 i}} \\ y(x)\to \left (\frac {1}{2}+\frac {i}{2}\right ) x^{\frac {3}{2}-\frac {3 i}{2}} e^{2 i \text {Interval}[\{0,\pi \}]} \sqrt {e^{2 i \text {Interval}[\{0,\pi \}]}-x^{6 i}} \\ \end{align*}