Internal problem ID [3343]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 21
Problem number: 601.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class A], _exact, _rational, _dAlembert]
Solve \begin {gather*} \boxed {\left (x^{2}-y^{2}\right ) y^{\prime }+x \left (x +2 y\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 419
dsolve((x^2-y(x)^2)*diff(y(x),x)+x*(x+2*y(x)) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\frac {\left (4+4 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {-3 x^{6} c_{1}^{3}+2 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}{2}+\frac {2 x^{2} c_{1}}{\left (4+4 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {-3 x^{6} c_{1}^{3}+2 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}}{\sqrt {c_{1}}} \\ y \relax (x ) = \frac {-\frac {\left (4+4 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {-3 x^{6} c_{1}^{3}+2 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}{4}-\frac {x^{2} c_{1}}{\left (4+4 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {-3 x^{6} c_{1}^{3}+2 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (4+4 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {-3 x^{6} c_{1}^{3}+2 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}{2}-\frac {2 x^{2} c_{1}}{\left (4+4 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {-3 x^{6} c_{1}^{3}+2 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}\right )}{2}}{\sqrt {c_{1}}} \\ y \relax (x ) = \frac {-\frac {\left (4+4 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {-3 x^{6} c_{1}^{3}+2 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}{4}-\frac {x^{2} c_{1}}{\left (4+4 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {-3 x^{6} c_{1}^{3}+2 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (4+4 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {-3 x^{6} c_{1}^{3}+2 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}{2}-\frac {2 x^{2} c_{1}}{\left (4+4 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {-3 x^{6} c_{1}^{3}+2 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}\right )}{2}}{\sqrt {c_{1}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 38.904 (sec). Leaf size: 611
DSolve[(x^2-y[x]^2)y'[x]+x(x+2 y[x])==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{x^3+\sqrt {-3 x^6+2 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}{\sqrt [3]{2}}+\frac {\sqrt [3]{2} x^2}{\sqrt [3]{x^3+\sqrt {-3 x^6+2 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}} \\ y(x)\to \frac {i \left (\sqrt [3]{2} \left (\sqrt {3}+i\right ) \left (x^3+\sqrt {-3 x^6+2 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}\right ){}^{2/3}-2 \left (\sqrt {3}-i\right ) x^2\right )}{2\ 2^{2/3} \sqrt [3]{x^3+\sqrt {-3 x^6+2 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}} \\ y(x)\to \frac {2 i \sqrt [3]{2} \left (\sqrt {3}+i\right ) x^2+2^{2/3} \left (-1-i \sqrt {3}\right ) \left (x^3+\sqrt {-3 x^6+2 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}\right ){}^{2/3}}{4 \sqrt [3]{x^3+\sqrt {-3 x^6+2 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}} \\ y(x)\to \frac {\sqrt [3]{\sqrt {3} \sqrt {-x^6}+x^3}}{\sqrt [3]{2}}+\frac {\sqrt [3]{2} x^2}{\sqrt [3]{\sqrt {3} \sqrt {-x^6}+x^3}} \\ y(x)\to \frac {\left (-2-2 i \sqrt {3}\right ) x^2+i \sqrt [3]{2} \left (\sqrt {3}+i\right ) \left (\sqrt {3} \sqrt {-x^6}+x^3\right )^{2/3}}{2\ 2^{2/3} \sqrt [3]{\sqrt {3} \sqrt {-x^6}+x^3}} \\ y(x)\to \frac {2 i \sqrt [3]{2} \left (\sqrt {3}+i\right ) x^2+2^{2/3} \left (-1-i \sqrt {3}\right ) \left (\sqrt {3} \sqrt {-x^6}+x^3\right )^{2/3}}{4 \sqrt [3]{\sqrt {3} \sqrt {-x^6}+x^3}} \\ \end{align*}