ODE
\[ 2 y(x) y''(x)=y(x)^2 (a+b y(x))+y'(x)^2 \] ODE Classification
[[_2nd_order, _missing_x]]
Book solution method
TO DO
Mathematica ✓
cpu = 1.8323 (sec), leaf count = 443
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [-\frac {2 i \sqrt {2} \text {$\#$1} \sqrt {\frac {\frac {2 c_1}{\text {$\#$1}}+\sqrt {a^2-2 b c_1}+a}{a+\sqrt {a^2-2 b c_1}}} \sqrt {1+\frac {2 c_1}{\text {$\#$1} \left (a-\sqrt {a^2-2 b c_1}\right )}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {c_1}{a+\sqrt {a^2-2 b c_1}}}}{\sqrt {\text {$\#$1}}}\right )|\frac {a+\sqrt {a^2-2 b c_1}}{a-\sqrt {a^2-2 b c_1}}\right )}{\sqrt {\frac {c_1}{a+\sqrt {a^2-2 b c_1}}} \sqrt {2 \text {$\#$1}^2 b+4 (\text {$\#$1} a+c_1)}}\& \right ][x+c_2]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {2 i \sqrt {2} \text {$\#$1} \sqrt {\frac {\frac {2 c_1}{\text {$\#$1}}+\sqrt {a^2-2 b c_1}+a}{a+\sqrt {a^2-2 b c_1}}} \sqrt {1+\frac {2 c_1}{\text {$\#$1} \left (a-\sqrt {a^2-2 b c_1}\right )}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {c_1}{a+\sqrt {a^2-2 b c_1}}}}{\sqrt {\text {$\#$1}}}\right )|\frac {a+\sqrt {a^2-2 b c_1}}{a-\sqrt {a^2-2 b c_1}}\right )}{\sqrt {\frac {c_1}{a+\sqrt {a^2-2 b c_1}}} \sqrt {2 \text {$\#$1}^2 b+4 (\text {$\#$1} a+c_1)}}\& \right ][x+c_2]\right \}\right \}\]
Maple ✓
cpu = 0.327 (sec), leaf count = 71
\[\left [\int _{}^{y \left (x \right )}-\frac {2}{\sqrt {2 b \,\textit {\_a}^{3}+4 a \,\textit {\_a}^{2}+4 \textit {\_a} \textit {\_C1}}}d \textit {\_a} -x -\textit {\_C2} = 0, \int _{}^{y \left (x \right )}\frac {2}{\sqrt {2 b \,\textit {\_a}^{3}+4 a \,\textit {\_a}^{2}+4 \textit {\_a} \textit {\_C1}}}d \textit {\_a} -x -\textit {\_C2} = 0\right ]\] Mathematica raw input
DSolve[2*y[x]*y''[x] == y[x]^2*(a + b*y[x]) + y'[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[((-2*I)*Sqrt[2]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[C[1]/(a + Sqrt[a^2 - 2*b*C[1]])])/Sqrt[#1]], (a + Sqrt[a^2 - 2*b*C[1]])/(a - Sqrt[a^2 - 2*b*C[1]])]*Sqrt[(a + Sqrt[a^2 - 2*b*C[1]] + (2*C[1])/#1)/(a + Sqrt[a^2 - 2*b*C[1]])]*Sqrt[1 + (2*C[1])/((a - Sqrt[a^2 - 2*b*C[1]])*#1)]*#1)/(Sqrt[C[1]/(a + Sqrt[a^2 - 2*b*C[1]])]*Sqrt[2*b*#1^2 + 4*(C[1] + a*#1)]) & ][x + C[2]]}, {y[x] -> InverseFunction[((2*I)*Sqrt[2]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[C[1]/(a + Sqrt[a^2 - 2*b*C[1]])])/Sqrt[#1]], (a + Sqrt[a^2 - 2*b*C[1]])/(a - Sqrt[a^2 - 2*b*C[1]])]*Sqrt[(a + Sqrt[a^2 - 2*b*C[1]] + (2*C[1])/#1)/(a + Sqrt[a^2 - 2*b*C[1]])]*Sqrt[1 + (2*C[1])/((a - Sqrt[a^2 - 2*b*C[1]])*#1)]*#1)/(Sqrt[C[1]/(a + Sqrt[a^2 - 2*b*C[1]])]*Sqrt[2*b*#1^2 + 4*(C[1] + a*#1)]) & ][x + C[2]]}}
Maple raw input
dsolve(2*y(x)*diff(diff(y(x),x),x) = diff(y(x),x)^2+(a+b*y(x))*y(x)^2, y(x))
Maple raw output
[Intat(-2/(2*_a^3*b+4*_a^2*a+4*_C1*_a)^(1/2),_a = y(x))-x-_C2 = 0, Intat(2/(2*_a^3*b+4*_a^2*a+4*_C1*_a)^(1/2),_a = y(x))-x-_C2 = 0]