4.39.45 $2y(x)y^{″}(x)=y^{\prime }(x{)}^2+8y(x{)}^3+4y(x{)}^2$

ODE
$$2y(x)y^{″}(x)=y^{\prime }(x{)}^2+8y(x{)}^3+4y(x{)}^2$$ ODE Classification

[[_2nd_order, _missing_x]]

Book solution method
TO DO

Mathematica ✓
cpu = 1.18291 (sec), leaf count = 359

$$\{\{y(x)\to \text{InverseFunction}[-\frac{2i\mathrm{\#}\text{1}\sqrt{\frac{c_1}{\mathrm{\#}\text{1}(2-2\sqrt{1-c_1})}+1}\sqrt{\frac{c_1}{\mathrm{\#}\text{1}(2\sqrt{1-c_1}+2)}+1}F(i{\sinh }^{-1}\left(\frac{\sqrt{\frac{c_1}{2\sqrt{1-c_1}+2}}}{\sqrt{\mathrm{\#}\text{1}}}\right)|\frac{\sqrt{1-c_1}+1}{1-\sqrt{1-c_1}})}{\sqrt{\frac{c_1}{2\sqrt{1-c_1}+2}}\sqrt{4{\mathrm{\#}\text{1}}^2+4\mathrm{\#}\text{1}+c_1}}\mathrm{\&}][c_2+x]\},\{y(x)\to \text{InverseFunction}\left[\frac{2i\mathrm{\#}\text{1}\sqrt{\frac{c_1}{\mathrm{\#}\text{1}(2-2\sqrt{1-c_1})}+1}\sqrt{\frac{c_1}{\mathrm{\#}\text{1}(2\sqrt{1-c_1}+2)}+1}F(i{\sinh }^{-1}\left(\frac{\sqrt{\frac{c_1}{2\sqrt{1-c_1}+2}}}{\sqrt{\mathrm{\#}\text{1}}}\right)|\frac{\sqrt{1-c_1}+1}{1-\sqrt{1-c_1}})}{\sqrt{\frac{c_1}{2\sqrt{1-c_1}+2}}\sqrt{4{\mathrm{\#}\text{1}}^2+4\mathrm{\#}\text{1}+c_1}}\mathrm{\&}\right][c_2+x]\}\}$$

Maple ✓
cpu = 0.075 (sec), leaf count = 61

$$\{{\int }^{y\left(x\right)}\frac{1}{\sqrt{4{\mathit{\_}\mathit{a}}^3+\mathit{\_}\mathit{a}\mathit{\_}\mathit{C}\mathit{1}+4{\mathit{\_}\mathit{a}}^2}}d\mathit{\_}\mathit{a}-x-\mathit{\_}\mathit{C}\mathit{2}=0,{\int }^{y\left(x\right)}-\frac{1}{\sqrt{(4{\mathit{\_}\mathit{a}}^2+\mathit{\_}\mathit{C}\mathit{1}+4\mathit{\_}\mathit{a})\mathit{\_}\mathit{a}}}d\mathit{\_}\mathit{a}-x-\mathit{\_}\mathit{C}\mathit{2}=0\}$$ Mathematica raw input

DSolve[2*y[x]*y''[x] == 4*y[x]^2 + 8*y[x]^3 + y'[x]^2,y[x],x]

Mathematica raw output

{{y[x] -> InverseFunction[((-2*I)*EllipticF[I*ArcSinh[Sqrt[C[1]/(2 + 2*Sqrt[1 - 
C[1]])]/Sqrt[#1]], (1 + Sqrt[1 - C[1]])/(1 - Sqrt[1 - C[1]])]*Sqrt[1 + C[1]/((2 
- 2*Sqrt[1 - C[1]])*#1)]*Sqrt[1 + C[1]/((2 + 2*Sqrt[1 - C[1]])*#1)]*#1)/(Sqrt[C[
1]/(2 + 2*Sqrt[1 - C[1]])]*Sqrt[C[1] + 4*#1 + 4*#1^2]) & ][x + C[2]]}, {y[x] -> 
InverseFunction[((2*I)*EllipticF[I*ArcSinh[Sqrt[C[1]/(2 + 2*Sqrt[1 - C[1]])]/Sqr
t[#1]], (1 + Sqrt[1 - C[1]])/(1 - Sqrt[1 - C[1]])]*Sqrt[1 + C[1]/((2 - 2*Sqrt[1 
- C[1]])*#1)]*Sqrt[1 + C[1]/((2 + 2*Sqrt[1 - C[1]])*#1)]*#1)/(Sqrt[C[1]/(2 + 2*S
qrt[1 - C[1]])]*Sqrt[C[1] + 4*#1 + 4*#1^2]) & ][x + C[2]]}}

Maple raw input

dsolve(2*y(x)*diff(diff(y(x),x),x) = diff(y(x),x)^2+4*y(x)^2+8*y(x)^3, y(x),'implicit')

Maple raw output

Intat(1/(4*_a^3+_C1*_a+4*_a^2)^(1/2),_a = y(x))-x-_C2 = 0, Intat(-1/((4*_a^2+_C1
+4*_a)*_a)^(1/2),_a = y(x))-x-_C2 = 0