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 = 0.579978 (sec), leaf count = 155
\[\left \{\text {Solve}\left [\frac {1}{4} \left (\frac {(a+2 b) \log \left (-\sqrt {a^2-\frac {4 b y(x)^2}{x^2}}+a+2 b\right )+a \log \left (\sqrt {a^2-\frac {4 b y(x)^2}{x^2}}+a\right )}{a+b}+2 \log (x)\right )=c_1,y(x)\right ],\text {Solve}\left [\frac {1}{4} \left (\frac {a \log \left (a-\sqrt {a^2-\frac {4 b y(x)^2}{x^2}}\right )+(a+2 b) \log \left (\sqrt {a^2-\frac {4 b y(x)^2}{x^2}}+a+2 b\right )}{a+b}+2 \log (x)\right )=c_1,y(x)\right ]\right \}\]
Maple ✓
cpu = 0.443 (sec), leaf count = 242
\[\left [\frac {\left (\frac {-a x +\sqrt {a^{2} x^{2}-4 b y \left (x \right )^{2}}}{2 y \left (x \right )}\right )^{-\frac {a}{a +b}} \left (-a x +\sqrt {a^{2} x^{2}-4 b y \left (x \right )^{2}}\right ) \left (-\frac {a \left (-a \,x^{2}+\sqrt {a^{2} x^{2}-4 b y \left (x \right )^{2}}\, x -2 y \left (x \right )^{2}\right )}{2 y \left (x \right )^{2}}\right )^{-\frac {a +2 b}{2 \left (a +b \right )}} x \textit {\_C1}}{y \left (x \right )^{2}}+x = 0, \frac {\left (a x +\sqrt {a^{2} x^{2}-4 b y \left (x \right )^{2}}\right ) \left (\frac {a \left (a \,x^{2}+\sqrt {a^{2} x^{2}-4 b y \left (x \right )^{2}}\, x +2 y \left (x \right )^{2}\right )}{2 y \left (x \right )^{2}}\right )^{-\frac {a +2 b}{2 \left (a +b \right )}} x \textit {\_C1} \left (-\frac {a x +\sqrt {a^{2} x^{2}-4 b y \left (x \right )^{2}}}{2 y \left (x \right )}\right )^{-\frac {a}{a +b}}}{y \left (x \right )^{2}}+x = 0\right ]\] Mathematica raw input
DSolve[b*y[x] + a*x*y'[x] + y[x]*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{Solve[(2*Log[x] + ((a + 2*b)*Log[a + 2*b - Sqrt[a^2 - (4*b*y[x]^2)/x^2]] + a*Log[a + Sqrt[a^2 - (4*b*y[x]^2)/x^2]])/(a + b))/4 == C[1], y[x]], Solve[(2*Log[x] + (a*Log[a - Sqrt[a^2 - (4*b*y[x]^2)/x^2]] + (a + 2*b)*Log[a + 2*b + Sqrt[a^2 - (4*b*y[x]^2)/x^2]])/(a + b))/4 == 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))
Maple raw output
[1/y(x)^2*(1/2/y(x)*(-a*x+(a^2*x^2-4*b*y(x)^2)^(1/2)))^(-a/(a+b))*(-a*x+(a^2*x^2-4*b*y(x)^2)^(1/2))*(-1/2*a*(-a*x^2+(a^2*x^2-4*b*y(x)^2)^(1/2)*x-2*y(x)^2)/y(x)^2)^(-1/2*(a+2*b)/(a+b))*x*_C1+x = 0, 1/y(x)^2*(a*x+(a^2*x^2-4*b*y(x)^2)^(1/2))*(1/2*a*(a*x^2+(a^2*x^2-4*b*y(x)^2)^(1/2)*x+2*y(x)^2)/y(x)^2)^(-1/2*(a+2*b)/(a+b))*x*_C1*(-1/2*(a*x+(a^2*x^2-4*b*y(x)^2)^(1/2))/y(x))^(-a/(a+b))+x = 0]