Internal problem ID [9151]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 816.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
\[ \boxed {y^{\prime }-\frac {\left (x -y\right )^{3} \left (x +y\right )^{3} x}{\left (-y^{2}+x^{2}-1\right ) y}=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 226
dsolve(diff(y(x),x) = (x-y(x))^3*(x+y(x))^3*x/(-y(x)^2+x^2-1)/y(x),y(x), singsol=all)
\[ -\left (\int _{\textit {\_b}}^{x}\frac {\left (-\textit {\_a} +y \left (x \right )\right )^{3} \left (\textit {\_a} +y \left (x \right )\right )^{3} \textit {\_a}}{-\textit {\_a}^{6}+3 \textit {\_a}^{4} y \left (x \right )^{2}-3 y \left (x \right )^{4} \textit {\_a}^{2}+y \left (x \right )^{6}+\textit {\_a}^{2}-y \left (x \right )^{2}-1}d \textit {\_a} \right )+\int _{}^{y \left (x \right )}\frac {2 \left (\left (-\textit {\_f}^{6}+3 \textit {\_f}^{4} x^{2}+\left (-3 x^{4}+1\right ) \textit {\_f}^{2}+x^{6}-x^{2}+1\right ) \left (\int _{\textit {\_b}}^{x}\frac {\left (\textit {\_a} -\textit {\_f} \right )^{2} \left (\textit {\_a} +\textit {\_f} \right )^{2} \textit {\_a} \left (2 \textit {\_a}^{2}-2 \textit {\_f}^{2}-3\right )}{\left (\textit {\_a}^{6}-3 \textit {\_a}^{4} \textit {\_f}^{2}+\left (3 \textit {\_f}^{4}-1\right ) \textit {\_a}^{2}-\textit {\_f}^{6}+\textit {\_f}^{2}+1\right )^{2}}d \textit {\_a} \right )+\frac {x^{2}}{2}-\frac {\textit {\_f}^{2}}{2}-\frac {1}{2}\right ) \textit {\_f}}{-\textit {\_f}^{6}+3 \textit {\_f}^{4} x^{2}+\left (-3 x^{4}+1\right ) \textit {\_f}^{2}+x^{6}-x^{2}+1}d \textit {\_f} +c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.193 (sec). Leaf size: 74
DSolve[y'[x] == (x*(x - y[x])^3*(x + y[x])^3)/(y[x]*(-1 + x^2 - y[x]^2)),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {1}{2} \left (\text {RootSum}\left [\text {$\#$1}^3-\text {$\#$1}+1\&,\frac {\text {$\#$1} \log \left (-\text {$\#$1}+x^2-y(x)^2\right )-\log \left (-\text {$\#$1}+x^2-y(x)^2\right )}{3 \text {$\#$1}^2-1}\&\right ]+x^2\right )=c_1,y(x)\right ] \]