y(x)y′′(x) = ay′(x)2 + by(x)3

4.39.25 $y (x) y^{″} (x) = a y^{'} (x)^{2} + b y (x)^{3}$

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

${{y (x) \to InverseFunction [- \frac{# 1 \sqrt{2 a - 3} \sqrt{1 - \frac{2 b {# 1}^{3 - 2 a}}{(2 a - 3) c_{1}}}_{2} F_{1} (\frac{1}{2}, \frac{a - 1}{2 a - 3}; \frac{a - 1}{2 a - 3} + 1; \frac{2 b {# 1}^{3 - 2 a}}{(2 a - 3) c_{1}})}{(a - 1) \sqrt{(2 a - 3) c_{1} {# 1}^{2 a} - 2 {# 1}^{3} b}} &] [c_{2} + x]}, {y (x) \to InverseFunction [\frac{# 1 \sqrt{2 a - 3} \sqrt{1 - \frac{2 b {# 1}^{3 - 2 a}}{(2 a - 3) c_{1}}}_{2} F_{1} (\frac{1}{2}, \frac{a - 1}{2 a - 3}; \frac{a - 1}{2 a - 3} + 1; \frac{2 b {# 1}^{3 - 2 a}}{(2 a - 3) c_{1}})}{(a - 1) \sqrt{(2 a - 3) c_{1} {# 1}^{2 a} - 2 {# 1}^{3} b}} &] [c_{2} + x]}}$

Maple ✓
cpu = 0.173 (sec), leaf count = 85

${\int^{y (x)} (- 2 a + 3) \frac{1}{\sqrt{(2 a - 3) (- 2 b {_a}^{3} +_C 1 {_a}^{2 a})}} d_a - x -_C 2 = 0, \int^{y (x)} (2 a - 3) \frac{1}{\sqrt{(2 a - 3) (- 2 b {_a}^{3} +_C 1 {_a}^{2 a})}} d_a - x -_C 2 = 0}$ 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]*Hyperg
eometric2F1[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, Int
at((-2*a+3)/((2*a-3)*(-2*b*_a^3+_C1*_a^(2*a)))^(1/2),_a = y(x))-x-_C2 = 0

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4.39.25 y(x)y″(x)=ay′(x)2+by(x)3

4.39.25 $y (x) y^{″} (x) = a y^{'} (x)^{2} + b y (x)^{3}$