ODE
\[ y(x) y''(x)=\text {a3} y(x)^{a+1}+a y'(x)^2+\text {a1} y(x) y'(x)+\text {a2} y(x)^2 \] ODE Classification
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 1.6153 (sec), leaf count = 370
\[\left \{\left \{y(x)\to \left (\frac {e^{-\frac {2 (a-1) \text {a2} x}{\sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}}-\text {a1}}} \left (\text {a1} \left (\text {a2} \left (c_2 \sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}}-(a-1) c_1 \exp \left (\frac {x \left (\text {a1} \sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}}+4 (a-1) \text {a2}-\text {a1}^2\right )}{\sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}}-\text {a1}}\right )\right )-\text {a3} \sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}} e^{\frac {2 (a-1) \text {a2} x}{\sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}}-\text {a1}}}\right )+(a-1) \text {a2} \left (-4 \text {a3} e^{\frac {2 (a-1) \text {a2} x}{\sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}}-\text {a1}}}+c_1 \sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}} \exp \left (\frac {x \left (\text {a1} \sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}}+4 (a-1) \text {a2}-\text {a1}^2\right )}{\sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}}-\text {a1}}\right )+4 \text {a2} c_2\right )+\text {a1}^2 \left (\text {a3} e^{\frac {2 (a-1) \text {a2} x}{\sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}}-\text {a1}}}-\text {a2} c_2\right )\right )}{\text {a2} \left (\text {a1} \sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}}+4 (a-1) \text {a2}-\text {a1}^2\right )}\right ){}^{\frac {1}{1-a}}\right \}\right \}\]
Maple ✓
cpu = 1.279 (sec), leaf count = 75
\[ \left \{ \sqrt { \left ( -4\,a+4 \right ) {\it a2}+{{\it a1}}^{2}} \left ( {\it a2}\, \left ( y \left ( x \right ) \right ) ^{1-a}+{\it a3} \right ) {{\rm e}^{{\frac {x}{2} \left ( -{\it a1}+\sqrt { \left ( -4\,a+4 \right ) {\it a2}+{{\it a1}}^{2}} \right ) }}}-{\it a2}\,{\it \_C1}\, \left ( a-1 \right ) {{\rm e}^{x\sqrt { \left ( -4\,a+4 \right ) {\it a2}+{{\it a1}}^{2}}}}+{\it \_C2}=0 \right \} \] Mathematica raw input
DSolve[y[x]*y''[x] == a2*y[x]^2 + a3*y[x]^(1 + a) + a1*y[x]*y'[x] + a*y'[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> ((a1^2*(a3*E^((2*(-1 + a)*a2*x)/(-a1 + Sqrt[a1^2 + 4*a2 - 4*a*a2])) - a2*C[2]) + (-1 + a)*a2*(-4*a3*E^((2*(-1 + a)*a2*x)/(-a1 + Sqrt[a1^2 + 4*a2 - 4*a*a2])) + Sqrt[a1^2 + 4*a2 - 4*a*a2]*E^(((-a1^2 + 4*(-1 + a)*a2 + a1*Sqrt[a1^2 + 4*a2 - 4*a*a2])*x)/(-a1 + Sqrt[a1^2 + 4*a2 - 4*a*a2]))*C[1] + 4*a2*C[2]) + a1*(-(Sqrt[a1^2 + 4*a2 - 4*a*a2]*a3*E^((2*(-1 + a)*a2*x)/(-a1 + Sqrt[a1^2 + 4*a2 - 4*a*a2]))) + a2*(-((-1 + a)*E^(((-a1^2 + 4*(-1 + a)*a2 + a1*Sqrt[a1^2 + 4*a2 - 4*a*a2])*x)/(-a1 + Sqrt[a1^2 + 4*a2 - 4*a*a2]))*C[1]) + Sqrt[a1^2 + 4*a2 - 4*a*a2]*C[2])))/(a2*(-a1^2 + 4*(-1 + a)*a2 + a1*Sqrt[a1^2 + 4*a2 - 4*a*a2])*E^((2*(-1 + a)*a2*x)/(-a1 + Sqrt[a1^2 + 4*a2 - 4*a*a2]))))^(1 - a)^(-1)}}
Maple raw input
dsolve(y(x)*diff(diff(y(x),x),x) = a*diff(y(x),x)^2+a1*y(x)*diff(y(x),x)+a2*y(x)^2+a3*y(x)^(1+a), y(x),'implicit')
Maple raw output
((-4*a+4)*a2+a1^2)^(1/2)*(a2*y(x)^(1-a)+a3)*exp(1/2*(-a1+((-4*a+4)*a2+a1^2)^(1/2))*x)-a2*_C1*(a-1)*exp(x*((-4*a+4)*a2+a1^2)^(1/2))+_C2 = 0