4.18.12 \(a y(x)+x y'(x)^2-y(x) y'(x)=0\)

ODE
\[ a y(x)+x y'(x)^2-y(x) y'(x)=0 \] ODE Classification

[[_homogeneous, `class A`], _rational, _dAlembert]

Book solution method
No Missing Variables ODE, Solve for \(x\)

Mathematica
cpu = 0.805737 (sec), leaf count = 158

\[\left \{\text {Solve}\left [\frac {y(x)}{a x}+\frac {\sqrt {\frac {y(x)}{x}} \sqrt {\frac {y(x)}{x}-4 a}}{a}+4 c_1+2 \log (x)=4 \log \left (\sqrt {\frac {y(x)}{x}-4 a}+\sqrt {\frac {y(x)}{x}}\right ),y(x)\right ],\text {Solve}\left [\frac {\sqrt {\frac {y(x)}{x}} \sqrt {\frac {y(x)}{x}-4 a}}{a}+4 c_1=\frac {y(x)}{a x}+4 \log \left (\sqrt {\frac {y(x)}{x}-4 a}+\sqrt {\frac {y(x)}{x}}\right )+2 \log (x),y(x)\right ]\right \}\]

Maple
cpu = 0.027 (sec), leaf count = 40

\[ \left \{ y \relax (x ) =0,[x \left ({\it \_T} \right ) ={ \left ({\it \_T}-a \right ) {\it \_C1} \left ({{\rm e}^{{\frac {{\it \_T}}{a}}}} \right ) ^{-1}},y \left ({\it \_T} \right ) ={{{\it \_T}}^{2}{\it \_C1} \left ({{\rm e}^{{\frac {{\it \_T}}{a}}}} \right ) ^{-1}}] \right \} \] Mathematica raw input

DSolve[a*y[x] - y[x]*y'[x] + x*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{Solve[4*C[1] + 2*Log[x] + y[x]/(a*x) + (Sqrt[y[x]/x]*Sqrt[-4*a + y[x]/x])/a == 
4*Log[Sqrt[y[x]/x] + Sqrt[-4*a + y[x]/x]], y[x]], Solve[4*C[1] + (Sqrt[y[x]/x]*S
qrt[-4*a + y[x]/x])/a == 2*Log[x] + 4*Log[Sqrt[y[x]/x] + Sqrt[-4*a + y[x]/x]] + 
y[x]/(a*x), y[x]]}

Maple raw input

dsolve(x*diff(y(x),x)^2-y(x)*diff(y(x),x)+a*y(x) = 0, y(x),'implicit')

Maple raw output

y(x) = 0, [x(_T) = (_T-a)*_C1/exp(_T/a), y(_T) = _T^2*_C1/exp(_T/a)]