ODE
\[ a y(x) y'(x)^2+b y(x)+y''(x)=0 \] ODE Classification
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.703747 (sec), leaf count = 96
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\sqrt {a}}{\sqrt {e^{2 a c_1-a K[1]^2}-b}}dK[1]\& \right ][x+c_2]\right \},\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt {a}}{\sqrt {e^{2 a c_1-a K[2]^2}-b}}dK[2]\& \right ][x+c_2]\right \}\right \}\]
Maple ✓
cpu = 0.821 (sec), leaf count = 70
\[\left [\int _{}^{y \left (x \right )}\frac {a}{\sqrt {a \left ({\mathrm e}^{-a \,\textit {\_a}^{2}} \textit {\_C1} a -b \right )}}d \textit {\_a} -x -\textit {\_C2} = 0, \int _{}^{y \left (x \right )}-\frac {a}{\sqrt {a \left ({\mathrm e}^{-a \,\textit {\_a}^{2}} \textit {\_C1} a -b \right )}}d \textit {\_a} -x -\textit {\_C2} = 0\right ]\] Mathematica raw input
DSolve[b*y[x] + a*y[x]*y'[x]^2 + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[Inactive[Integrate][-(Sqrt[a]/Sqrt[-b + E^(2*a*C[1] - a*K[1]^2)]), {K[1], 1, #1}] & ][x + C[2]]}, {y[x] -> InverseFunction[Inactive[Integrate][Sqrt[a]/Sqrt[-b + E^(2*a*C[1] - a*K[2]^2)], {K[2], 1, #1}] & ][x + C[2]]}}
Maple raw input
dsolve(diff(diff(y(x),x),x)+a*y(x)*diff(y(x),x)^2+b*y(x) = 0, y(x))
Maple raw output
[Intat(a/(a*(exp(-a*_a^2)*_C1*a-b))^(1/2),_a = y(x))-x-_C2 = 0, Intat(-a/(a*(exp(-a*_a^2)*_C1*a-b))^(1/2),_a = y(x))-x-_C2 = 0]