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 = 1.6976 (sec), leaf count = 539
\[\left \{\left \{y(x)\to 2^{\frac {1}{1-a}} \left (-\frac {e^{-\frac {2 (a-1) \text {a2} x}{\sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}}-\text {a1}}} \left (\text {a1}^2 \left (\text {a3} \left (-\sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}}\right ) e^{\frac {2 (a-1) \text {a2} x}{\sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}}-\text {a1}}}-(a-1) \text {a2} c_1 \exp \left (\frac {x \left (\text {a1} \sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}}+4 (a-1) \text {a2}-\text {a1}^2\right )}{\sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}}-\text {a1}}\right )+\text {a2} c_2 \sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}}\right )+(a-1) \text {a1} \text {a2} \left (-4 \text {a3} e^{\frac {2 (a-1) \text {a2} x}{\sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}}-\text {a1}}}+c_1 \sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}} \exp \left (\frac {x \left (\text {a1} \sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}}+4 (a-1) \text {a2}-\text {a1}^2\right )}{\sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}}-\text {a1}}\right )+4 \text {a2} c_2\right )+2 (a-1) \text {a2} \left (\text {a3} \sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}} e^{\frac {2 (a-1) \text {a2} x}{\sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}}-\text {a1}}}+(a-1) \text {a2} c_1 \exp \left (\frac {x \left (\text {a1} \sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}}+4 (a-1) \text {a2}-\text {a1}^2\right )}{\sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}}-\text {a1}}\right )-\text {a2} c_2 \sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}}\right )+\text {a1}^3 \left (\text {a3} e^{\frac {2 (a-1) \text {a2} x}{\sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}}-\text {a1}}}-\text {a2} c_2\right )\right )}{\text {a2} \left (\sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}}-\text {a1}\right ) \left (\text {a1} \sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}}+4 (a-1) \text {a2}-\text {a1}^2\right )}\right ){}^{\frac {1}{1-a}}\right \}\right \}\]
Maple ✓
cpu = 1.565 (sec), leaf count = 147
\[\left [y \left (x \right ) = {\mathrm e}^{\frac {x \sqrt {-4 \mathit {a2} a +\mathit {a1}^{2}+4 \mathit {a2}}}{-2+2 a}} {\mathrm e}^{-\frac {\mathit {a1} x}{2 \left (a -1\right )}} \left (\frac {\mathit {a2}^{2} \left (-4 \mathit {a2} a +\mathit {a1}^{2}+4 \mathit {a2} \right )}{\left (-{\mathrm e}^{x \sqrt {-4 \mathit {a2} a +\mathit {a1}^{2}+4 \mathit {a2}}} \textit {\_C1} a \mathit {a2} +\mathit {a3} \,{\mathrm e}^{\frac {\left (-\mathit {a1} +\sqrt {-4 \mathit {a2} a +\mathit {a1}^{2}+4 \mathit {a2}}\right ) x}{2}} \sqrt {-4 \mathit {a2} a +\mathit {a1}^{2}+4 \mathit {a2}}+{\mathrm e}^{x \sqrt {-4 \mathit {a2} a +\mathit {a1}^{2}+4 \mathit {a2}}} \textit {\_C1} \mathit {a2} +\mathit {a2} \textit {\_C2} a -\mathit {a2} \textit {\_C2} \right )^{2}}\right )^{\frac {1}{-2+2 a}}\right ]\] 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] -> 2^(1 - a)^(-1)*(-((a1^3*(a3*E^((2*(-1 + a)*a2*x)/(-a1 + Sqrt[a1^2 + 4*a2 - 4*a*a2])) - a2*C[2]) + (-1 + a)*a1*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]) + 2*(-1 + a)*a2*(Sqrt[a1^2 + 4*a2 - 4*a*a2]*a3*E^((2*(-1 + a)*a2*x)/(-a1 + Sqrt[a1^2 + 4*a2 - 4*a*a2])) + (-1 + 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] - a2*Sqrt[a1^2 + 4*a2 - 4*a*a2]*C[2]) + a1^2*(-(Sqrt[a1^2 + 4*a2 - 4*a*a2]*a3*E^((2*(-1 + a)*a2*x)/(-a1 + Sqrt[a1^2 + 4*a2 - 4*a*a2]))) - (-1 + 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] + a2*Sqrt[a1^2 + 4*a2 - 4*a*a2]*C[2]))/(a2*(-a1 + Sqrt[a1^2 + 4*a2 - 4*a*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))
Maple raw output
[y(x) = exp(1/2/(a-1)*x*(-4*a*a2+a1^2+4*a2)^(1/2))*exp(-1/2/(a-1)*a1*x)*(1/(-exp(x*(-4*a*a2+a1^2+4*a2)^(1/2))*_C1*a*a2+a3*exp(1/2*(-a1+(-4*a*a2+a1^2+4*a2)^(1/2))*x)*(-4*a*a2+a1^2+4*a2)^(1/2)+exp(x*(-4*a*a2+a1^2+4*a2)^(1/2))*_C1*a2+a2*_C2*a-a2*_C2)^2*a2^2*(-4*a*a2+a1^2+4*a2))^(1/2/(a-1))]