Internal problem ID [7996]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 416.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class A], _dAlembert]
Solve \begin {gather*} \boxed {x \left (y^{\prime }\right )^{2}+\left (y-3 x \right ) y^{\prime }+y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 136
dsolve(x*diff(y(x),x)^2+(y(x)-3*x)*diff(y(x),x)+y(x) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = x \\ -\frac {c_{1} \left (5 x -y \relax (x )+\sqrt {9 x^{2}-10 x y \relax (x )+y \relax (x )^{2}}\right )}{x \left (\frac {3 x -y \relax (x )+\sqrt {9 x^{2}-10 x y \relax (x )+y \relax (x )^{2}}}{x}\right )^{\frac {3}{2}}}+x = 0 \\ \frac {\left (-5 x +y \relax (x )+\sqrt {9 x^{2}-10 x y \relax (x )+y \relax (x )^{2}}\right ) c_{1}}{x \left (\frac {6 x -2 y \relax (x )-2 \sqrt {9 x^{2}-10 x y \relax (x )+y \relax (x )^{2}}}{x}\right )^{\frac {3}{2}}}+x = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.316 (sec). Leaf size: 1225
DSolve[y[x] + (-3*x + y[x])*y'[x] + x*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{384} \left (\frac {\frac {4 e^{8 c_1}}{x^2}-6912 e^{4 c_1}}{\sqrt [3]{-\frac {-373248 e^{4 c_1} x^4+4320 e^{8 c_1} x^2-48 \sqrt {6} x \sqrt {e^{8 c_1} \left (216 x^2+e^{4 c_1}\right ){}^3}+e^{12 c_1}}{x^3}}}+4 \sqrt [3]{\frac {373248 e^{4 c_1} x^4-4320 e^{8 c_1} x^2+48 \sqrt {6} x \sqrt {e^{8 c_1} \left (216 x^2+e^{4 c_1}\right ){}^3}-e^{12 c_1}}{x^3}}-\frac {4 e^{4 c_1}}{x}\right ) \\ y(x)\to \frac {1}{768} \left (\frac {\left (1+i \sqrt {3}\right ) \left (6912 e^{4 c_1}-\frac {4 e^{8 c_1}}{x^2}\right )}{\sqrt [3]{-\frac {-373248 e^{4 c_1} x^4+4320 e^{8 c_1} x^2-48 \sqrt {6} x \sqrt {e^{8 c_1} \left (216 x^2+e^{4 c_1}\right ){}^3}+e^{12 c_1}}{x^3}}}+4 i \left (\sqrt {3}+i\right ) \sqrt [3]{-\frac {-373248 e^{4 c_1} x^4+4320 e^{8 c_1} x^2-48 \sqrt {6} x \sqrt {e^{8 c_1} \left (216 x^2+e^{4 c_1}\right ){}^3}+e^{12 c_1}}{x^3}}-\frac {8 e^{4 c_1}}{x}\right ) \\ y(x)\to \frac {1}{768} \left (\frac {\left (1-i \sqrt {3}\right ) \left (6912 e^{4 c_1}-\frac {4 e^{8 c_1}}{x^2}\right )}{\sqrt [3]{-\frac {-373248 e^{4 c_1} x^4+4320 e^{8 c_1} x^2-48 \sqrt {6} x \sqrt {e^{8 c_1} \left (216 x^2+e^{4 c_1}\right ){}^3}+e^{12 c_1}}{x^3}}}-4 \left (1+i \sqrt {3}\right ) \sqrt [3]{-\frac {-373248 e^{4 c_1} x^4+4320 e^{8 c_1} x^2-48 \sqrt {6} x \sqrt {e^{8 c_1} \left (216 x^2+e^{4 c_1}\right ){}^3}+e^{12 c_1}}{x^3}}-\frac {8 e^{4 c_1}}{x}\right ) \\ y(x)\to \frac {1}{3} \left (\frac {4-108 e^{4 c_1} x^2}{x^2 \sqrt [3]{\frac {729 e^{20 c_1} x^4-540 e^{16 c_1} x^2+3 \sqrt {3} x \sqrt {e^{28 c_1} \left (8+27 e^{4 c_1} x^2\right ){}^3}-8 e^{12 c_1}}{x^3}}}+e^{-8 c_1} \sqrt [3]{\frac {729 e^{20 c_1} x^4-540 e^{16 c_1} x^2+3 \sqrt {3} x \sqrt {e^{28 c_1} \left (8+27 e^{4 c_1} x^2\right ){}^3}-8 e^{12 c_1}}{x^3}}-\frac {2 e^{-4 c_1}}{x}\right ) \\ y(x)\to \frac {1}{6} e^{-8 c_1} \left (\frac {\left (1+i \sqrt {3}\right ) \left (108 e^{12 c_1}-\frac {4 e^{8 c_1}}{x^2}\right )}{\sqrt [3]{\frac {729 e^{20 c_1} x^4-540 e^{16 c_1} x^2+3 \sqrt {3} x \sqrt {e^{28 c_1} \left (8+27 e^{4 c_1} x^2\right ){}^3}-8 e^{12 c_1}}{x^3}}}+i \left (\sqrt {3}+i\right ) \sqrt [3]{\frac {729 e^{20 c_1} x^4-540 e^{16 c_1} x^2+3 \sqrt {3} x \sqrt {e^{28 c_1} \left (8+27 e^{4 c_1} x^2\right ){}^3}-8 e^{12 c_1}}{x^3}}-\frac {4 e^{4 c_1}}{x}\right ) \\ y(x)\to \frac {1}{6} e^{-8 c_1} \left (\frac {\left (1-i \sqrt {3}\right ) \left (108 e^{12 c_1}-\frac {4 e^{8 c_1}}{x^2}\right )}{\sqrt [3]{\frac {729 e^{20 c_1} x^4-540 e^{16 c_1} x^2+3 \sqrt {3} x \sqrt {e^{28 c_1} \left (8+27 e^{4 c_1} x^2\right ){}^3}-8 e^{12 c_1}}{x^3}}}-\left (1+i \sqrt {3}\right ) \sqrt [3]{\frac {729 e^{20 c_1} x^4-540 e^{16 c_1} x^2+3 \sqrt {3} x \sqrt {e^{28 c_1} \left (8+27 e^{4 c_1} x^2\right ){}^3}-8 e^{12 c_1}}{x^3}}-\frac {4 e^{4 c_1}}{x}\right ) \\ \end{align*}