Internal problem ID [3361]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 22
Problem number: 621.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class A], _exact, _rational, _dAlembert]
Solve \begin {gather*} \boxed {\left (2 x^{2}+4 y x -y^{2}\right ) y^{\prime }-x^{2}+4 y x +2 y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 441
dsolve((2*x^2+4*x*y(x)-y(x)^2)*diff(y(x),x) = x^2-4*x*y(x)-2*y(x)^2,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\frac {\left (108 x^{3} c_{1}^{3}+4+4 \sqrt {-135 x^{6} c_{1}^{6}+54 x^{3} c_{1}^{3}+1}\right )^{\frac {1}{3}}}{2}+\frac {12 x^{2} c_{1}^{2}}{\left (108 x^{3} c_{1}^{3}+4+4 \sqrt {-135 x^{6} c_{1}^{6}+54 x^{3} c_{1}^{3}+1}\right )^{\frac {1}{3}}}+2 c_{1} x}{c_{1}} \\ y \relax (x ) = \frac {-\frac {\left (108 x^{3} c_{1}^{3}+4+4 \sqrt {-135 x^{6} c_{1}^{6}+54 x^{3} c_{1}^{3}+1}\right )^{\frac {1}{3}}}{4}-\frac {6 x^{2} c_{1}^{2}}{\left (108 x^{3} c_{1}^{3}+4+4 \sqrt {-135 x^{6} c_{1}^{6}+54 x^{3} c_{1}^{3}+1}\right )^{\frac {1}{3}}}+2 c_{1} x -\frac {i \sqrt {3}\, \left (\frac {\left (108 x^{3} c_{1}^{3}+4+4 \sqrt {-135 x^{6} c_{1}^{6}+54 x^{3} c_{1}^{3}+1}\right )^{\frac {1}{3}}}{2}-\frac {12 x^{2} c_{1}^{2}}{\left (108 x^{3} c_{1}^{3}+4+4 \sqrt {-135 x^{6} c_{1}^{6}+54 x^{3} c_{1}^{3}+1}\right )^{\frac {1}{3}}}\right )}{2}}{c_{1}} \\ y \relax (x ) = \frac {-\frac {\left (108 x^{3} c_{1}^{3}+4+4 \sqrt {-135 x^{6} c_{1}^{6}+54 x^{3} c_{1}^{3}+1}\right )^{\frac {1}{3}}}{4}-\frac {6 x^{2} c_{1}^{2}}{\left (108 x^{3} c_{1}^{3}+4+4 \sqrt {-135 x^{6} c_{1}^{6}+54 x^{3} c_{1}^{3}+1}\right )^{\frac {1}{3}}}+2 c_{1} x +\frac {i \sqrt {3}\, \left (\frac {\left (108 x^{3} c_{1}^{3}+4+4 \sqrt {-135 x^{6} c_{1}^{6}+54 x^{3} c_{1}^{3}+1}\right )^{\frac {1}{3}}}{2}-\frac {12 x^{2} c_{1}^{2}}{\left (108 x^{3} c_{1}^{3}+4+4 \sqrt {-135 x^{6} c_{1}^{6}+54 x^{3} c_{1}^{3}+1}\right )^{\frac {1}{3}}}\right )}{2}}{c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 25.959 (sec). Leaf size: 670
DSolve[(2 x^2+4 x y[x]-y[x]^2)y'[x]==x^2-4 x y[x]-2 y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{27 x^3+\sqrt {-135 x^6+54 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}{\sqrt [3]{2}}+\frac {6 \sqrt [3]{2} x^2}{\sqrt [3]{27 x^3+\sqrt {-135 x^6+54 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}+2 x \\ y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{27 x^3+\sqrt {-135 x^6+54 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}{2 \sqrt [3]{2}}-\frac {3 \sqrt [3]{2} \left (1+i \sqrt {3}\right ) x^2}{\sqrt [3]{27 x^3+\sqrt {-135 x^6+54 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}+2 x \\ y(x)\to -\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{27 x^3+\sqrt {-135 x^6+54 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}{2 \sqrt [3]{2}}+\frac {3 i \sqrt [3]{2} \left (\sqrt {3}+i\right ) x^2}{\sqrt [3]{27 x^3+\sqrt {-135 x^6+54 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}+2 x \\ y(x)\to \sqrt [3]{\frac {3}{2}} \sqrt [3]{\sqrt {15} \sqrt {-x^6}+9 x^3}+\frac {2 \sqrt [3]{2} 3^{2/3} x^2}{\sqrt [3]{\sqrt {15} \sqrt {-x^6}+9 x^3}}+2 x \\ y(x)\to -\frac {1}{2} i \sqrt [3]{\frac {3}{2}} \left (\sqrt {3}-i\right ) \sqrt [3]{\sqrt {15} \sqrt {-x^6}+9 x^3}-\frac {\sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}-3 i\right ) x^2}{\sqrt [3]{\sqrt {15} \sqrt {-x^6}+9 x^3}}+2 x \\ y(x)\to \frac {1}{2} i \sqrt [3]{\frac {3}{2}} \left (\sqrt {3}+i\right ) \sqrt [3]{\sqrt {15} \sqrt {-x^6}+9 x^3}-\frac {\sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}+3 i\right ) x^2}{\sqrt [3]{\sqrt {15} \sqrt {-x^6}+9 x^3}}+2 x \\ \end{align*}