Internal problem ID [7972]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 392.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(y)]]]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{2}-x y^{\prime } y+y^{2} \ln \left (a y\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.362 (sec). Leaf size: 50
dsolve(diff(y(x),x)^2-x*y(x)*diff(y(x),x)+y(x)^2*ln(a*y(x)) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {{\mathrm e}^{\frac {x^{2}}{4}}}{a} \\ y \relax (x ) = \frac {{\mathrm e}^{-c_{1}^{2}} {\mathrm e}^{c_{1} x}}{a} \\ y \relax (x ) = \frac {{\mathrm e}^{-c_{1}^{2}} {\mathrm e}^{-c_{1} x}}{a} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.318 (sec). Leaf size: 30
DSolve[Log[a*y[x]]*y[x]^2 - x*y[x]*y'[x] + y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{\frac {1}{4} c_1 (2 x-c_1)}}{a} \\ y(x)\to 0 \\ \end{align*}