ODE
\[ y(x) y''(x)=a y'(x)^2+b y(x)^3 \] ODE Classification
[[_2nd_order, _missing_x]]
Book solution method
TO DO
Mathematica ✓
cpu = 2.17783 (sec), leaf count = 264
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [-\frac {\text {$\#$1} \sqrt {2 a-3} \sqrt {1-\frac {2 b \text {$\#$1}^{3-2 a}}{(2 a-3) c_1}} \, _2F_1\left (\frac {1}{2},\frac {a-1}{2 a-3};\frac {a-1}{2 a-3}+1;\frac {2 b \text {$\#$1}^{3-2 a}}{(2 a-3) c_1}\right )}{(a-1) \sqrt {(2 a-3) c_1 \text {$\#$1}^{2 a}-2 \text {$\#$1}^3 b}}\& \right ]\left [c_2+x\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {\text {$\#$1} \sqrt {2 a-3} \sqrt {1-\frac {2 b \text {$\#$1}^{3-2 a}}{(2 a-3) c_1}} \, _2F_1\left (\frac {1}{2},\frac {a-1}{2 a-3};\frac {a-1}{2 a-3}+1;\frac {2 b \text {$\#$1}^{3-2 a}}{(2 a-3) c_1}\right )}{(a-1) \sqrt {(2 a-3) c_1 \text {$\#$1}^{2 a}-2 \text {$\#$1}^3 b}}\& \right ]\left [c_2+x\right ]\right \}\right \}\]
Maple ✓
cpu = 0.173 (sec), leaf count = 85
\[ \left \{ \int ^{y \left ( x \right ) }\!{(-2\,a+3){\frac {1}{\sqrt { \left ( 2\,a-3 \right ) \left ( -2\,b{{\it \_a}}^{3}+{\it \_C1}\,{{\it \_a}}^{2\,a} \right ) }}}}{d{\it \_a}}-x-{\it \_C2}=0,\int ^{y \left ( x \right ) }\!{(2\,a-3){\frac {1}{\sqrt { \left ( 2\,a-3 \right ) \left ( -2\,b{{\it \_a}}^{3}+{\it \_C1}\,{{\it \_a}}^{2\,a} \right ) }}}}{d{\it \_a}}-x-{\it \_C2}=0 \right \} \] Mathematica raw input
DSolve[y[x]*y''[x] == b*y[x]^3 + a*y'[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[-((Sqrt[-3 + 2*a]*Hypergeometric2F1[1/2, (-1 + a)/(-3 + 2*a), 1 + (-1 + a)/(-3 + 2*a), (2*b*#1^(3 - 2*a))/((-3 + 2*a)*C[1])]*#1*Sqrt[1 - (2*b*#1^(3 - 2*a))/((-3 + 2*a)*C[1])])/((-1 + a)*Sqrt[-2*b*#1^3 + (-3 + 2*a)*C[1]*#1^(2*a)])) & ][x + C[2]]}, {y[x] -> InverseFunction[(Sqrt[-3 + 2*a]*Hypergeometric2F1[1/2, (-1 + a)/(-3 + 2*a), 1 + (-1 + a)/(-3 + 2*a), (2*b*#1^(3 - 2*a))/((-3 + 2*a)*C[1])]*#1*Sqrt[1 - (2*b*#1^(3 - 2*a))/((-3 + 2*a)*C[1])])/((-1 + a)*Sqrt[-2*b*#1^3 + (-3 + 2*a)*C[1]*#1^(2*a)]) & ][x + C[2]]}}
Maple raw input
dsolve(y(x)*diff(diff(y(x),x),x) = a*diff(y(x),x)^2+b*y(x)^3, y(x),'implicit')
Maple raw output
Intat((2*a-3)/((2*a-3)*(-2*b*_a^3+_C1*_a^(2*a)))^(1/2),_a = y(x))-x-_C2 = 0, Intat((-2*a+3)/((2*a-3)*(-2*b*_a^3+_C1*_a^(2*a)))^(1/2),_a = y(x))-x-_C2 = 0