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

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

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

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

Mathematica
cpu = 0.635193 (sec), leaf count = 243

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

Maple
cpu = 0.023 (sec), leaf count = 106

\[ \left \{ [x \left ({\it \_T} \right ) ={{\it \_T}}^{{\frac {1}{a} \left (1-{a}^{-1} \right ) ^{-1}}} \left ({\frac {b}{{{\it \_T}}^{ \left (a-1 \right ) ^{-1}} \left (2\,a-1 \right ) {{\it \_T}}^{2}}}+{\it \_C1} \right ) ,y \left ({\it \_T} \right ) ={\frac {1}{{\it \_T}\,a{{\it \_T}}^{ \left (a-1 \right ) ^{-1}} \left (2\,a-1 \right ) } \left (2\, \left ({\it \_C1}\,{{\it \_T}}^{ \left (a-1 \right ) ^{-1}}{{\it \_T}}^{2}+b \right ) \left (a-1/2 \right ) {{\it \_T}}^{ \left (a-1 \right ) ^{-1}}+{{\it \_T}}^{ \left (a-1 \right ) ^{-1}}b \right ) }] \right \} \] Mathematica raw input

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

Mathematica raw output

{Solve[(-2*a*ArcTanh[Sqrt[-4*b*x + a^2*y[x]^2]/(a*y[x])] - 2*(-1 + a)*ArcTanh[Sq
rt[-4*b*x + a^2*y[x]^2]/(y[x] - a*y[x])] + a*Log[4*b*x] - Log[4*b*x + (1 - 2*a)*
y[x]^2] + a*Log[4*b*x + (1 - 2*a)*y[x]^2])/(-1 + 2*a) == C[1], y[x]], Solve[(2*a
*ArcTanh[Sqrt[-4*b*x + a^2*y[x]^2]/(a*y[x])] + 2*(-1 + a)*ArcTanh[Sqrt[-4*b*x + 
a^2*y[x]^2]/(y[x] - a*y[x])] + a*Log[4*b*x] - Log[4*b*x + (1 - 2*a)*y[x]^2] + a*
Log[4*b*x + (1 - 2*a)*y[x]^2])/(-1 + 2*a) == C[1], y[x]]}

Maple raw input

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

Maple raw output

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