4.41.32 $2y(x{)}^3{y}^{″}(x)+y(x{)}^2{y}^{\prime }(x{)}^2=2$

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

[[_2nd_order, _missing_x]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.664688 (sec), leaf count = 721

$$\{\{y(x)\to \frac{c_1^2(3\sqrt{\frac{(c_2+x){}^2(9c_1^4{x}^2+18c_2{c}_1^{4}x+9c_2^2{c}_1^4+128)}{c_1^2}}+9c_1(c_2+x){}^2+\frac{64}{c_1^3}){}^{2/3}-4c_1\sqrt[3]{3\sqrt{\frac{(c_2+x){}^2(9c_1^4{x}^2+18c_2{c}_1^{4}x+9c_2^2{c}_1^4+128)}{c_1^2}}+9c_1(c_2+x){}^2+\frac{64}{c_1^3}}+16}{2c_1^2\sqrt[3]{3\sqrt{\frac{(c_2+x){}^2(9c_1^4{x}^2+18c_2{c}_1^{4}x+9c_2^2{c}_1^4+128)}{c_1^2}}+9c_1(c_2+x){}^2+\frac{64}{c_1^3}}}\},\{y(x)\to \frac{i(c_1\sqrt[3]{3\sqrt{\frac{(c_2+x){}^2(9c_1^4{x}^2+18c_2{c}_1^{4}x+9c_2^2{c}_1^4+128)}{c_1^2}}+9c_1(c_2+x){}^2+\frac{64}{c_1^3}}+4)((\sqrt{3}+i)c_1\sqrt[3]{3\sqrt{\frac{(c_2+x){}^2(9c_1^4{x}^2+18c_2{c}_1^{4}x+9c_2^2{c}_1^4+128)}{c_1^2}}+9c_1(c_2+x){}^2+\frac{64}{c_1^3}}-4\sqrt{3}+4i)}{4c_1^2\sqrt[3]{3\sqrt{\frac{(c_2+x){}^2(9c_1^4{x}^2+18c_2{c}_1^{4}x+9c_2^2{c}_1^4+128)}{c_1^2}}+9c_1(c_2+x){}^2+\frac{64}{c_1^3}}}\},\{y(x)\to -\frac{i(c_1\sqrt[3]{3\sqrt{\frac{(c_2+x){}^2(9c_1^4{x}^2+18c_2{c}_1^{4}x+9c_2^2{c}_1^4+128)}{c_1^2}}+9c_1(c_2+x){}^2+\frac{64}{c_1^3}}+4)((\sqrt{3}-i)c_1\sqrt[3]{3\sqrt{\frac{(c_2+x){}^2(9c_1^4{x}^2+18c_2{c}_1^{4}x+9c_2^2{c}_1^4+128)}{c_1^2}}+9c_1(c_2+x){}^2+\frac{64}{c_1^3}}-4(\sqrt{3}+i))}{4c_1^2\sqrt[3]{3\sqrt{\frac{(c_2+x){}^2(9c_1^4{x}^2+18c_2{c}_1^{4}x+9c_2^2{c}_1^4+128)}{c_1^2}}+9c_1(c_2+x){}^2+\frac{64}{c_1^3}}}\}\}$$

Maple ✓
cpu = 0.084 (sec), leaf count = 57

$$\{-\frac{2\mathit{\_}\mathit{C}\mathit{1}y\left(x\right)+8}{3{\mathit{\_}\mathit{C}\mathit{1}}^2}\sqrt{\mathit{\_}\mathit{C}\mathit{1}y\left(x\right)-2}-x-\mathit{\_}\mathit{C}\mathit{2}=0,\frac{2\mathit{\_}\mathit{C}\mathit{1}y\left(x\right)+8}{3{\mathit{\_}\mathit{C}\mathit{1}}^2}\sqrt{\mathit{\_}\mathit{C}\mathit{1}y\left(x\right)-2}-x-\mathit{\_}\mathit{C}\mathit{2}=0\}$$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> (16 - 4*C[1]*(64/C[1]^3 + 9*C[1]*(x + C[2])^2 + 3*Sqrt[((x + C[2])^2*(
128 + 9*x^2*C[1]^4 + 18*x*C[1]^4*C[2] + 9*C[1]^4*C[2]^2))/C[1]^2])^(1/3) + C[1]^
2*(64/C[1]^3 + 9*C[1]*(x + C[2])^2 + 3*Sqrt[((x + C[2])^2*(128 + 9*x^2*C[1]^4 + 
18*x*C[1]^4*C[2] + 9*C[1]^4*C[2]^2))/C[1]^2])^(2/3))/(2*C[1]^2*(64/C[1]^3 + 9*C[
1]*(x + C[2])^2 + 3*Sqrt[((x + C[2])^2*(128 + 9*x^2*C[1]^4 + 18*x*C[1]^4*C[2] + 
9*C[1]^4*C[2]^2))/C[1]^2])^(1/3))}, {y[x] -> ((I/4)*(4 + C[1]*(64/C[1]^3 + 9*C[1
]*(x + C[2])^2 + 3*Sqrt[((x + C[2])^2*(128 + 9*x^2*C[1]^4 + 18*x*C[1]^4*C[2] + 9
*C[1]^4*C[2]^2))/C[1]^2])^(1/3))*(4*I - 4*Sqrt[3] + (I + Sqrt[3])*C[1]*(64/C[1]^
3 + 9*C[1]*(x + C[2])^2 + 3*Sqrt[((x + C[2])^2*(128 + 9*x^2*C[1]^4 + 18*x*C[1]^4
*C[2] + 9*C[1]^4*C[2]^2))/C[1]^2])^(1/3)))/(C[1]^2*(64/C[1]^3 + 9*C[1]*(x + C[2]
)^2 + 3*Sqrt[((x + C[2])^2*(128 + 9*x^2*C[1]^4 + 18*x*C[1]^4*C[2] + 9*C[1]^4*C[2
]^2))/C[1]^2])^(1/3))}, {y[x] -> ((-I/4)*(4 + C[1]*(64/C[1]^3 + 9*C[1]*(x + C[2]
)^2 + 3*Sqrt[((x + C[2])^2*(128 + 9*x^2*C[1]^4 + 18*x*C[1]^4*C[2] + 9*C[1]^4*C[2
]^2))/C[1]^2])^(1/3))*(-4*(I + Sqrt[3]) + (-I + Sqrt[3])*C[1]*(64/C[1]^3 + 9*C[1
]*(x + C[2])^2 + 3*Sqrt[((x + C[2])^2*(128 + 9*x^2*C[1]^4 + 18*x*C[1]^4*C[2] + 9
*C[1]^4*C[2]^2))/C[1]^2])^(1/3)))/(C[1]^2*(64/C[1]^3 + 9*C[1]*(x + C[2])^2 + 3*S
qrt[((x + C[2])^2*(128 + 9*x^2*C[1]^4 + 18*x*C[1]^4*C[2] + 9*C[1]^4*C[2]^2))/C[1
]^2])^(1/3))}}

Maple raw input

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

Maple raw output

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