Internal problem ID [8396]
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]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {\left (x -y\right )^{3} \left (x +y\right )^{3} x}{\left (-y^{2}+x^{2}-1\right ) y}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 307
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)
\[ \int _{\textit {\_b}}^{x}\frac {\left (\textit {\_a} -y \relax (x )\right )^{3} \left (\textit {\_a} +y \relax (x )\right )^{3} \textit {\_a}}{\textit {\_a}^{6}-3 \textit {\_a}^{4} y \relax (x )^{2}+3 \textit {\_a}^{2} y \relax (x )^{4}-y \relax (x )^{6}-\textit {\_a}^{2}+y \relax (x )^{2}+1}d \textit {\_a} +\int _{}^{y \relax (x )}\left (-\frac {\left (-\textit {\_f}^{2}+x^{2}-1\right ) \textit {\_f}}{-\textit {\_f}^{6}+3 \textit {\_f}^{4} x^{2}-3 \textit {\_f}^{2} x^{4}+x^{6}+\textit {\_f}^{2}-x^{2}+1}-\left (\int _{\textit {\_b}}^{x}\left (-\frac {\left (\textit {\_a} -\textit {\_f} \right )^{3} \left (\textit {\_a} +\textit {\_f} \right )^{3} \textit {\_a} \left (-6 \textit {\_a}^{4} \textit {\_f} +12 \textit {\_a}^{2} \textit {\_f}^{3}-6 \textit {\_f}^{5}+2 \textit {\_f} \right )}{\left (\textit {\_a}^{6}-3 \textit {\_a}^{4} \textit {\_f}^{2}+3 \textit {\_a}^{2} \textit {\_f}^{4}-\textit {\_f}^{6}-\textit {\_a}^{2}+\textit {\_f}^{2}+1\right )^{2}}-\frac {3 \left (\textit {\_a} -\textit {\_f} \right )^{2} \left (\textit {\_a} +\textit {\_f} \right )^{3} \textit {\_a}}{\textit {\_a}^{6}-3 \textit {\_a}^{4} \textit {\_f}^{2}+3 \textit {\_a}^{2} \textit {\_f}^{4}-\textit {\_f}^{6}-\textit {\_a}^{2}+\textit {\_f}^{2}+1}+\frac {3 \left (\textit {\_a} -\textit {\_f} \right )^{3} \left (\textit {\_a} +\textit {\_f} \right )^{2} \textit {\_a}}{\textit {\_a}^{6}-3 \textit {\_a}^{4} \textit {\_f}^{2}+3 \textit {\_a}^{2} \textit {\_f}^{4}-\textit {\_f}^{6}-\textit {\_a}^{2}+\textit {\_f}^{2}+1}\right )d \textit {\_a} \right )\right )d \textit {\_f} +c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.182 (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]
\[ \text {Solve}\left [\frac {1}{2} \left (\text {RootSum}\left [\text {$\#$1}^3-\text {$\#$1}+1\&,\frac {\text {$\#$1} \log \left (-\text {$\#$1}+x^2-y(x)^2\right )-\log \left (-\text {$\#$1}+x^2-y(x)^2\right )}{3 \text {$\#$1}^2-1}\&\right ]+x^2\right )=c_1,y(x)\right ] \]