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 = 0.242426 (sec), leaf count = 25
\[\left \{\left \{y(x)\to \frac {e^{\frac {1}{4} c_1 \left (2 x-c_1\right )}}{a}\right \}\right \}\]
Maple ✓
cpu = 0.273 (sec), leaf count = 56
\[ \left \{ -{\it \_C1}+x-\sqrt {{x}^{2}-4\,\ln \left ( ay \left ( x \right ) \right ) }=0,-{\it \_C1}+\sqrt {{x}^{2}-4\,\ln \left ( ay \left ( x \right ) \right ) }+x=0,y \left ( x \right ) ={\frac {1}{a}{{\rm e}^{{\frac {{x}^{2}}{4}}}}} \right \} \] 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),'implicit')
Maple raw output
y(x) = exp(1/4*x^2)/a, -_C1+x-(x^2-4*ln(a*y(x)))^(1/2) = 0, -_C1+(x^2-4*ln(a*y(x)))^(1/2)+x = 0