Internal problem ID [7980]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 400.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class G]]
Solve \begin {gather*} \boxed {2 \left (y^{\prime }\right )^{2}-2 y^{\prime } x^{2}+3 y x=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 109
dsolve(2*diff(y(x),x)^2-2*x^2*diff(y(x),x)+3*x*y(x) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {x^{3}}{6} \\ y \relax (x ) = \frac {x^{3}}{3}-\frac {\left (x^{2}-\sqrt {-6 x c_{1}}\right ) x}{3}+c_{1} \\ y \relax (x ) = \frac {x^{3}}{3}-\frac {\left (x^{2}+\sqrt {-6 x c_{1}}\right ) x}{3}+c_{1} \\ y \relax (x ) = \frac {x^{3}}{3}+\frac {\left (-x^{2}-\sqrt {-6 x c_{1}}\right ) x}{3}+c_{1} \\ y \relax (x ) = \frac {x^{3}}{3}+\frac {\left (-x^{2}+\sqrt {-6 x c_{1}}\right ) x}{3}+c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 3.596 (sec). Leaf size: 334
DSolve[3*x*y[x] - 2*x^2*y'[x] + 2*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\int _1^{y(x)}\left (\frac {\sqrt {x^3-6 K[1]} \sqrt {x \left (x^3-6 K[1]\right )}}{3 x^{7/2} K[1]}-\frac {2 \sqrt {x^3-6 K[1]} \sqrt {x \left (x^3-6 K[1]\right )}}{x^{7/2} \left (6 K[1]-x^3\right )}+\frac {1}{3 K[1]}\right )dK[1]-\frac {2 \sqrt {x^4-6 x y(x)} \log \left (x^{3/2}+\sqrt {x^3-6 y(x)}\right )}{3 \sqrt {x} \sqrt {x^3-6 y(x)}}=c_1,y(x)\right ] \\ \text {Solve}\left [\int _1^{y(x)}\left (-\frac {\sqrt {x^3-6 K[2]} \sqrt {x \left (x^3-6 K[2]\right )}}{3 x^{7/2} K[2]}+\frac {2 \sqrt {x^3-6 K[2]} \sqrt {x \left (x^3-6 K[2]\right )}}{x^{7/2} \left (6 K[2]-x^3\right )}+\frac {1}{3 K[2]}\right )dK[2]+\frac {2 \sqrt {x^4-6 x y(x)} \log \left (x^{3/2}+\sqrt {x^3-6 y(x)}\right )}{3 \sqrt {x} \sqrt {x^3-6 y(x)}}=c_1,y(x)\right ] \\ y(x)\to \frac {x^3}{6} \\ \end{align*}