ODE
\[ y(x) y''(x)=a y'(x)^2+b y(x) y'(x)+c y(x)^2 \] ODE Classification
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.527405 (sec), leaf count = 51
\[\left \{\left \{y(x)\to c_2 \exp \left (-\frac {b x+2 \log \left (\cos \left (\frac {1}{2} (x+c_1) \sqrt {4 (a-1) c-b^2}\right )\right )}{2 (a-1)}\right )\right \}\right \}\]
Maple ✓
cpu = 1.216 (sec), leaf count = 103
\[\left [y \left (x \right ) = {\mathrm e}^{\frac {x \sqrt {-4 a c +b^{2}+4 c}}{-2+2 a}} {\mathrm e}^{-\frac {b x}{2 \left (a -1\right )}} \left (\frac {-4 a c +b^{2}+4 c}{\left ({\mathrm e}^{x \sqrt {-4 a c +b^{2}+4 c}} \textit {\_C1} a -{\mathrm e}^{x \sqrt {-4 a c +b^{2}+4 c}} \textit {\_C1} -\textit {\_C2} a +\textit {\_C2} \right )^{2}}\right )^{\frac {1}{-2+2 a}}\right ]\] Mathematica raw input
DSolve[y[x]*y''[x] == c*y[x]^2 + b*y[x]*y'[x] + a*y'[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> C[2]/E^((b*x + 2*Log[Cos[(Sqrt[-b^2 + 4*(-1 + a)*c]*(x + C[1]))/2]])/(2*(-1 + a)))}}
Maple raw input
dsolve(y(x)*diff(diff(y(x),x),x) = a*diff(y(x),x)^2+b*y(x)*diff(y(x),x)+c*y(x)^2, y(x))
Maple raw output
[y(x) = exp(1/2/(a-1)*x*(-4*a*c+b^2+4*c)^(1/2))*exp(-1/2/(a-1)*b*x)*(1/(exp(x*(-4*a*c+b^2+4*c)^(1/2))*_C1*a-exp(x*(-4*a*c+b^2+4*c)^(1/2))*_C1-_C2*a+_C2)^2*(-4*a*c+b^2+4*c))^(1/2/(a-1))]