ODE
\[ y(x)^2 \log (a y(x))-x y(x) y'(x)+y'(x)^2=0 \] ODE Classification
[[_1st_order, `_with_symmetry_[F(x),G(y)]`]]
Book solution method
No Missing Variables ODE, Solve for \(x\)
Mathematica ✓
cpu = 0.39159 (sec), leaf count = 25
\[\left \{\left \{y(x)\to \frac {e^{\frac {1}{4} c_1 (2 x-c_1)}}{a}\right \}\right \}\]
Maple ✓
cpu = 4.008 (sec), leaf count = 50
\[\left [y \left (x \right ) = \frac {{\mathrm e}^{\frac {x^{2}}{4}}}{a}, y \left (x \right ) = \frac {{\mathrm e}^{-\textit {\_C1}^{2}} {\mathrm e}^{x \textit {\_C1}}}{a}, y \left (x \right ) = \frac {{\mathrm e}^{-\textit {\_C1}^{2}} {\mathrm e}^{-x \textit {\_C1}}}{a}\right ]\] Mathematica raw input
DSolve[Log[a*y[x]]*y[x]^2 - x*y[x]*y'[x] + y'[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> E^(((2*x - C[1])*C[1])/4)/a}}
Maple raw input
dsolve(diff(y(x),x)^2-x*y(x)*diff(y(x),x)+y(x)^2*ln(a*y(x)) = 0, y(x))
Maple raw output
[y(x) = exp(1/4*x^2)/a, y(x) = 1/exp(_C1^2)*exp(x*_C1)/a, y(x) = 1/exp(_C1^2)/exp(x*_C1)/a]