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], _rational, _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: 0.242 (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 (-y \relax (x )+5 x +\sqrt {9 x^{2}-10 x y \relax (x )+y \relax (x )^{2}}\right )}{x \left (\frac {-y \relax (x )+3 x +\sqrt {9 x^{2}-10 x y \relax (x )+y \relax (x )^{2}}}{x}\right )^{\frac {3}{2}}}+x = 0 \\ \frac {\left (y \relax (x )-5 x +\sqrt {9 x^{2}-10 x y \relax (x )+y \relax (x )^{2}}\right ) c_{1}}{x \left (\frac {-2 y \relax (x )+6 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: 0.265 (sec). Leaf size: 1248
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 {x^2 \left (\frac {-23887872 e^{4 c_1} x^4-34560 e^{8 c_1} x^2+192 \sqrt {3} x \sqrt {-e^{8 c_1} \left (-1728 x^2+e^{4 c_1}\right ){}^3}+e^{12 c_1}}{x^3}\right ){}^{2/3}+e^{4 c_1} x \left (13824 x+\sqrt [3]{\frac {-23887872 e^{4 c_1} x^4-34560 e^{8 c_1} x^2+192 \sqrt {3} x \sqrt {-e^{8 c_1} \left (-1728 x^2+e^{4 c_1}\right ){}^3}+e^{12 c_1}}{x^3}}\right )+e^{8 c_1}}{768 x^2 \sqrt [3]{\frac {-23887872 e^{4 c_1} x^4-34560 e^{8 c_1} x^2+192 \sqrt {3} x \sqrt {-e^{8 c_1} \left (-1728 x^2+e^{4 c_1}\right ){}^3}+e^{12 c_1}}{x^3}}} \\ y(x)\to \frac {\frac {\left (-1-i \sqrt {3}\right ) e^{4 c_1} \left (13824 x^2+e^{4 c_1}\right )}{x^2 \sqrt [3]{\frac {-23887872 e^{4 c_1} x^4-34560 e^{8 c_1} x^2+192 \sqrt {3} x \sqrt {-e^{8 c_1} \left (-1728 x^2+e^{4 c_1}\right ){}^3}+e^{12 c_1}}{x^3}}}+i \left (\sqrt {3}+i\right ) \sqrt [3]{\frac {-23887872 e^{4 c_1} x^4-34560 e^{8 c_1} x^2+192 \sqrt {3} x \sqrt {-e^{8 c_1} \left (-1728 x^2+e^{4 c_1}\right ){}^3}+e^{12 c_1}}{x^3}}+\frac {2 e^{4 c_1}}{x}}{1536} \\ y(x)\to \frac {\frac {i \left (\sqrt {3}+i\right ) e^{4 c_1} \left (13824 x^2+e^{4 c_1}\right )}{x^2 \sqrt [3]{\frac {-23887872 e^{4 c_1} x^4-34560 e^{8 c_1} x^2+192 \sqrt {3} x \sqrt {-e^{8 c_1} \left (-1728 x^2+e^{4 c_1}\right ){}^3}+e^{12 c_1}}{x^3}}}-\left (1+i \sqrt {3}\right ) \sqrt [3]{\frac {-23887872 e^{4 c_1} x^4-34560 e^{8 c_1} x^2+192 \sqrt {3} x \sqrt {-e^{8 c_1} \left (-1728 x^2+e^{4 c_1}\right ){}^3}+e^{12 c_1}}{x^3}}+\frac {2 e^{4 c_1}}{x}}{1536} \\ y(x)\to \frac {\frac {-4+6912 e^{4 c_1} x^2}{\sqrt [3]{-373248 e^{20 c_1} x^4+4320 e^{16 c_1} x^2+48 \sqrt {6} \sqrt {e^{28 c_1} x^2 \left (1+216 e^{4 c_1} x^2\right ){}^3}+e^{12 c_1}}}-4 e^{-8 c_1} \sqrt [3]{-373248 e^{20 c_1} x^4+4320 e^{16 c_1} x^2+48 \sqrt {6} \sqrt {e^{28 c_1} x^2 \left (1+216 e^{4 c_1} x^2\right ){}^3}+e^{12 c_1}}-4 e^{-4 c_1}}{384 x} \\ y(x)\to \frac {\frac {\left (-4-4 i \sqrt {3}\right ) \left (-1+1728 e^{4 c_1} x^2\right )}{\sqrt [3]{-373248 e^{20 c_1} x^4+4320 e^{16 c_1} x^2+48 \sqrt {6} \sqrt {e^{28 c_1} x^2 \left (1+216 e^{4 c_1} x^2\right ){}^3}+e^{12 c_1}}}+\left (4-4 i \sqrt {3}\right ) e^{-8 c_1} \sqrt [3]{-373248 e^{20 c_1} x^4+4320 e^{16 c_1} x^2+48 \sqrt {6} \sqrt {e^{28 c_1} x^2 \left (1+216 e^{4 c_1} x^2\right ){}^3}+e^{12 c_1}}-8 e^{-4 c_1}}{768 x} \\ y(x)\to \frac {\frac {4 i \left (\sqrt {3}+i\right ) \left (-1+1728 e^{4 c_1} x^2\right )}{\sqrt [3]{-373248 e^{20 c_1} x^4+4320 e^{16 c_1} x^2+48 \sqrt {6} \sqrt {e^{28 c_1} x^2 \left (1+216 e^{4 c_1} x^2\right ){}^3}+e^{12 c_1}}}+\left (4+4 i \sqrt {3}\right ) e^{-8 c_1} \sqrt [3]{-373248 e^{20 c_1} x^4+4320 e^{16 c_1} x^2+48 \sqrt {6} \sqrt {e^{28 c_1} x^2 \left (1+216 e^{4 c_1} x^2\right ){}^3}+e^{12 c_1}}-8 e^{-4 c_1}}{768 x} \\ \end{align*}