Internal problem ID [3388]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 23
Problem number: 649.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class A], _rational, _dAlembert]
Solve \begin {gather*} \boxed {x \left (x^{2}-2 y^{2}\right ) y^{\prime }-\left (2 x^{2}-y^{2}\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 807
dsolve(x*(x^2-2*y(x)^2)*diff(y(x),x) = (2*x^2-y(x)^2)*y(x),y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {216 x^{2} c_{1}}{\left (6 \left (8-108 x^{2} c_{1}^{2}+12 \sqrt {81 x^{4} c_{1}^{4}-12 x^{2} c_{1}^{2}}\right )^{\frac {1}{3}}+\frac {24}{\left (8-108 x^{2} c_{1}^{2}+12 \sqrt {81 x^{4} c_{1}^{4}-12 x^{2} c_{1}^{2}}\right )^{\frac {1}{3}}}+12\right )^{\frac {3}{2}}} \\ y \relax (x ) = \frac {216 x^{2} c_{1}}{\left (6 \left (8-108 x^{2} c_{1}^{2}+12 \sqrt {81 x^{4} c_{1}^{4}-12 x^{2} c_{1}^{2}}\right )^{\frac {1}{3}}+\frac {24}{\left (8-108 x^{2} c_{1}^{2}+12 \sqrt {81 x^{4} c_{1}^{4}-12 x^{2} c_{1}^{2}}\right )^{\frac {1}{3}}}+12\right )^{\frac {3}{2}}} \\ y \relax (x ) = -\frac {216 x^{2} c_{1}}{\left (-3 \left (8-108 x^{2} c_{1}^{2}+12 \sqrt {81 x^{4} c_{1}^{4}-12 x^{2} c_{1}^{2}}\right )^{\frac {1}{3}}-\frac {12}{\left (8-108 x^{2} c_{1}^{2}+12 \sqrt {81 x^{4} c_{1}^{4}-12 x^{2} c_{1}^{2}}\right )^{\frac {1}{3}}}+12-18 i \sqrt {3}\, \left (\frac {\left (8-108 x^{2} c_{1}^{2}+12 \sqrt {81 x^{4} c_{1}^{4}-12 x^{2} c_{1}^{2}}\right )^{\frac {1}{3}}}{6}-\frac {2}{3 \left (8-108 x^{2} c_{1}^{2}+12 \sqrt {81 x^{4} c_{1}^{4}-12 x^{2} c_{1}^{2}}\right )^{\frac {1}{3}}}\right )\right )^{\frac {3}{2}}} \\ y \relax (x ) = \frac {216 x^{2} c_{1}}{\left (-3 \left (8-108 x^{2} c_{1}^{2}+12 \sqrt {81 x^{4} c_{1}^{4}-12 x^{2} c_{1}^{2}}\right )^{\frac {1}{3}}-\frac {12}{\left (8-108 x^{2} c_{1}^{2}+12 \sqrt {81 x^{4} c_{1}^{4}-12 x^{2} c_{1}^{2}}\right )^{\frac {1}{3}}}+12-18 i \sqrt {3}\, \left (\frac {\left (8-108 x^{2} c_{1}^{2}+12 \sqrt {81 x^{4} c_{1}^{4}-12 x^{2} c_{1}^{2}}\right )^{\frac {1}{3}}}{6}-\frac {2}{3 \left (8-108 x^{2} c_{1}^{2}+12 \sqrt {81 x^{4} c_{1}^{4}-12 x^{2} c_{1}^{2}}\right )^{\frac {1}{3}}}\right )\right )^{\frac {3}{2}}} \\ y \relax (x ) = -\frac {216 x^{2} c_{1}}{\left (-3 \left (8-108 x^{2} c_{1}^{2}+12 \sqrt {81 x^{4} c_{1}^{4}-12 x^{2} c_{1}^{2}}\right )^{\frac {1}{3}}-\frac {12}{\left (8-108 x^{2} c_{1}^{2}+12 \sqrt {81 x^{4} c_{1}^{4}-12 x^{2} c_{1}^{2}}\right )^{\frac {1}{3}}}+12+18 i \sqrt {3}\, \left (\frac {\left (8-108 x^{2} c_{1}^{2}+12 \sqrt {81 x^{4} c_{1}^{4}-12 x^{2} c_{1}^{2}}\right )^{\frac {1}{3}}}{6}-\frac {2}{3 \left (8-108 x^{2} c_{1}^{2}+12 \sqrt {81 x^{4} c_{1}^{4}-12 x^{2} c_{1}^{2}}\right )^{\frac {1}{3}}}\right )\right )^{\frac {3}{2}}} \\ y \relax (x ) = \frac {216 x^{2} c_{1}}{\left (-3 \left (8-108 x^{2} c_{1}^{2}+12 \sqrt {81 x^{4} c_{1}^{4}-12 x^{2} c_{1}^{2}}\right )^{\frac {1}{3}}-\frac {12}{\left (8-108 x^{2} c_{1}^{2}+12 \sqrt {81 x^{4} c_{1}^{4}-12 x^{2} c_{1}^{2}}\right )^{\frac {1}{3}}}+12+18 i \sqrt {3}\, \left (\frac {\left (8-108 x^{2} c_{1}^{2}+12 \sqrt {81 x^{4} c_{1}^{4}-12 x^{2} c_{1}^{2}}\right )^{\frac {1}{3}}}{6}-\frac {2}{3 \left (8-108 x^{2} c_{1}^{2}+12 \sqrt {81 x^{4} c_{1}^{4}-12 x^{2} c_{1}^{2}}\right )^{\frac {1}{3}}}\right )\right )^{\frac {3}{2}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.275 (sec). Leaf size: 831
DSolve[x(x^2-2 y[x]^2)y'[x]==(2 x^2-y[x]^2)y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {-x^2+\frac {\sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}{\sqrt [3]{2} 3^{2/3}}+\frac {\sqrt [3]{\frac {2}{3}} e^{2 c_1} x^2}{\sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}} \\ y(x)\to \sqrt {-x^2+\frac {\sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}{\sqrt [3]{2} 3^{2/3}}+\frac {\sqrt [3]{\frac {2}{3}} e^{2 c_1} x^2}{\sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}} \\ y(x)\to -\frac {1}{2} \sqrt {\left (\frac {2}{3}\right )^{2/3} \left (-1-i \sqrt {3}\right ) \sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}+\frac {4}{3} x^2 \left (-3+\frac {(-3)^{2/3} \sqrt [3]{2} e^{2 c_1}}{\sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}\right )} \\ y(x)\to \frac {1}{2} \sqrt {\left (\frac {2}{3}\right )^{2/3} \left (-1-i \sqrt {3}\right ) \sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}+\frac {4}{3} x^2 \left (-3+\frac {(-3)^{2/3} \sqrt [3]{2} e^{2 c_1}}{\sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}\right )} \\ y(x)\to -\frac {1}{2} \sqrt {-4 x^2+i \left (\frac {2}{3}\right )^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}-\frac {4 \sqrt [3]{-\frac {2}{3}} e^{2 c_1} x^2}{\sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}} \\ y(x)\to \frac {1}{2} \sqrt {-4 x^2+i \left (\frac {2}{3}\right )^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}-\frac {4 \sqrt [3]{-\frac {2}{3}} e^{2 c_1} x^2}{\sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}} \\ \end{align*}