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 = 0.346277 (sec), leaf count = 51
\[\left \{\left \{y(x)\to c_2 \exp \left (-\frac {2 \log \left (\cos \left (\frac {1}{2} \left (c_1+x\right ) \sqrt {4 (a-1) c-b^2}\right )\right )+b x}{2 (a-1)}\right )\right \}\right \}\]
Maple ✓
cpu = 0.13 (sec), leaf count = 70
\[ \left \{ \left ( y \left ( x \right ) \right ) ^{1-a}{{\rm e}^{{\frac {x}{2} \left ( -b+\sqrt { \left ( -4\,a+4 \right ) c+{b}^{2}} \right ) }}}\sqrt { \left ( -4\,a+4 \right ) c+{b}^{2}}-{\it \_C1}\, \left ( a-1 \right ) {{\rm e}^{x\sqrt { \left ( -4\,a+4 \right ) c+{b}^{2}}}}+{\it \_C2}=0 \right \} \] 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),'implicit')
Maple raw output
y(x)^(1-a)*exp(1/2*(-b+((-4*a+4)*c+b^2)^(1/2))*x)*((-4*a+4)*c+b^2)^(1/2)-_C1*(a-1)*exp(x*((-4*a+4)*c+b^2)^(1/2))+_C2 = 0