Internal problem ID [3833]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 21
Problem number: 583.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]
\[ \boxed {\left (1-y x^{3}\right ) y^{\prime }-x^{2} y^{2}=0} \]
✓ Solution by Maple
Time used: 0.36 (sec). Leaf size: 789
dsolve((1-x^3*y(x))*diff(y(x),x) = x^2*y(x)^2,y(x), singsol=all)
\begin{align*} y \left (x \right ) = \frac {{\left (\frac {\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{c_{1}}-\frac {c_{1}}{\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}\right )}^{2}+3}{2 x^{3}} y \left (x \right ) = \frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{6} {\left (\frac {\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{c_{1}}-\frac {c_{1}}{\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}\right )}^{2}+3}{2 x^{3}} y \left (x \right ) = \frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{6} {\left (\frac {\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{c_{1}}-\frac {c_{1}}{\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}\right )}^{2}+3}{2 x^{3}} y \left (x \right ) = \frac {\frac {{\left (-\frac {4 \left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{c_{1}}+\frac {4 c_{1}}{\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}-4 i \sqrt {3}\, \left (\frac {\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{c_{1}}+\frac {c_{1}}{\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}\right )\right )}^{2}}{64}+3}{2 x^{3}} y \left (x \right ) = \frac {\frac {{\left (-\frac {4 \left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{c_{1}}+\frac {4 c_{1}}{\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}+4 i \sqrt {3}\, \left (\frac {\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{c_{1}}+\frac {c_{1}}{\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}\right )\right )}^{2}}{64}+3}{2 x^{3}} y \left (x \right ) = \frac {\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{6} {\left (-\frac {4 \left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{c_{1}}+\frac {4 c_{1}}{\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}-4 i \sqrt {3}\, \left (\frac {\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{c_{1}}+\frac {c_{1}}{\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}\right )\right )}^{2}}{64}+3}{2 x^{3}} y \left (x \right ) = \frac {\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{6} {\left (-\frac {4 \left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{c_{1}}+\frac {4 c_{1}}{\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}+4 i \sqrt {3}\, \left (\frac {\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{c_{1}}+\frac {c_{1}}{\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}\right )\right )}^{2}}{64}+3}{2 x^{3}} y \left (x \right ) = \frac {\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{6} {\left (-\frac {4 \left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{c_{1}}+\frac {4 c_{1}}{\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}-4 i \sqrt {3}\, \left (\frac {\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{c_{1}}+\frac {c_{1}}{\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}\right )\right )}^{2}}{64}+3}{2 x^{3}} y \left (x \right ) = \frac {\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{6} {\left (-\frac {4 \left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{c_{1}}+\frac {4 c_{1}}{\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}+4 i \sqrt {3}\, \left (\frac {\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{c_{1}}+\frac {c_{1}}{\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}\right )\right )}^{2}}{64}+3}{2 x^{3}} \end{align*}
✓ Solution by Mathematica
Time used: 50.23 (sec). Leaf size: 331
DSolve[(1-x^3 y[x])y'[x]==x^2 y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{12 c_1 x^6+2 \sqrt {6} \sqrt {c_1 x^6 \left (1+6 c_1 x^6\right )}+1}+\frac {1}{\sqrt [3]{12 c_1 x^6+2 \sqrt {6} \sqrt {c_1 x^6 \left (1+6 c_1 x^6\right )}+1}}+1}{2 x^3} y(x)\to \frac {2 i \left (\sqrt {3}+i\right ) \sqrt [3]{12 c_1 x^6+2 \sqrt {6} \sqrt {c_1 x^6 \left (1+6 c_1 x^6\right )}+1}-\frac {2 \left (1+i \sqrt {3}\right )}{\sqrt [3]{12 c_1 x^6+2 \sqrt {6} \sqrt {c_1 x^6 \left (1+6 c_1 x^6\right )}+1}}+4}{8 x^3} y(x)\to \frac {-2 \left (1+i \sqrt {3}\right ) \sqrt [3]{12 c_1 x^6+2 \sqrt {6} \sqrt {c_1 x^6 \left (1+6 c_1 x^6\right )}+1}+\frac {2 i \left (\sqrt {3}+i\right )}{\sqrt [3]{12 c_1 x^6+2 \sqrt {6} \sqrt {c_1 x^6 \left (1+6 c_1 x^6\right )}+1}}+4}{8 x^3} y(x)\to 0 \end{align*}