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

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

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

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

Mathematica ✓
cpu = 0.467269 (sec), leaf count = 245

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

Maple ✓
cpu = 0.057 (sec), leaf count = 112

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

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

Mathematica raw output

{Solve[(4*Log[x] + (2*a*ArcTanh[Sqrt[a^2 - (4*b*y[x]^2)/x^2]/a] - 2*(a + 2*b)*Ar
cTanh[Sqrt[a^2 - (4*b*y[x]^2)/x^2]/(a + 2*b)] + a*Log[y[x]^2/x^2] + a*Log[a + b 
+ y[x]^2/x^2] + 2*b*Log[a + b + y[x]^2/x^2])/(a + b))/8 == C[1], y[x]], Solve[(4
*Log[x] + (-2*a*ArcTanh[Sqrt[a^2 - (4*b*y[x]^2)/x^2]/a] + 2*(a + 2*b)*ArcTanh[Sq
rt[a^2 - (4*b*y[x]^2)/x^2]/(a + 2*b)] + a*Log[y[x]^2/x^2] + a*Log[a + b + y[x]^2
/x^2] + 2*b*Log[a + b + y[x]^2/x^2])/(a + b))/8 == C[1], y[x]]}

Maple raw input

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

Maple raw output

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