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

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

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

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

Mathematica ✓
cpu = 0.24869 (sec), leaf count = 167

\[\left \{\text {Solve}\left [\frac {4 a \tan ^{-1}\left (\frac {y(x)}{x \sqrt {4 a-\frac {y(x)^2}{x^2}}}\right )+\frac {y(x) \left (\sqrt {4 a-\frac {y(x)^2}{x^2}}-\frac {i y(x)}{x}\right )}{x}}{8 a}=c_1+\frac {1}{2} i \log (x),y(x)\right ],\text {Solve}\left [\frac {4 a x^2 \left (\tan ^{-1}\left (\frac {y(x)}{x \sqrt {4 a-\frac {y(x)^2}{x^2}}}\right )-2 c_1+i \log (x)\right )+x y(x) \sqrt {4 a-\frac {y(x)^2}{x^2}}+i y(x)^2}{a x}=0,y(x)\right ]\right \}\]

Maple ✓
cpu = 0.028 (sec), leaf count = 36

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

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

Mathematica raw output

{Solve[(4*a*ArcTan[y[x]/(x*Sqrt[4*a - y[x]^2/x^2])] + (y[x]*(((-I)*y[x])/x + Sqr
t[4*a - y[x]^2/x^2]))/x)/(8*a) == C[1] + (I/2)*Log[x], y[x]], Solve[(4*a*x^2*(Ar
cTan[y[x]/(x*Sqrt[4*a - y[x]^2/x^2])] - 2*C[1] + I*Log[x]) + I*y[x]^2 + x*y[x]*S
qrt[4*a - y[x]^2/x^2])/(a*x) == 0, y[x]]}

Maple raw input

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

Maple raw output

[x(_T) = exp(-1/2/a*_T^2)*_T*_C1, y(_T) = (_T^2+a)*exp(-1/2/a*_T^2)*_C1]