4.40.9 $2y(x)y^{″}(x)=y^{\prime }(x{)}^2(y^{\prime }(x{)}^2+1)$

ODE
$$2y(x)y^{″}(x)=y^{\prime }(x{)}^2(y^{\prime }(x{)}^2+1)$$ ODE Classification

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.908026 (sec), leaf count = 173

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

Maple ✓
cpu = 0.078 (sec), leaf count = 87

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

DSolve[2*y[x]*y''[x] == y'[x]^2*(1 + y'[x]^2),y[x],x]

Mathematica raw output

{{y[x] -> InverseFunction[((-I)*(-(Log[E^(2*C[1])*Sqrt[#1] + E^C[1]*Sqrt[-1 + E^
(2*C[1])*#1]]/E^C[1]) + Sqrt[#1]*Sqrt[-1 + E^(2*C[1])*#1]))/E^C[1] & ][x + C[2]]
}, {y[x] -> InverseFunction[(I*(-(Log[E^(2*C[1])*Sqrt[#1] + E^C[1]*Sqrt[-1 + E^(
2*C[1])*#1]]/E^C[1]) + Sqrt[#1]*Sqrt[-1 + E^(2*C[1])*#1]))/E^C[1] & ][x + C[2]]}
}

Maple raw input

dsolve(2*y(x)*diff(diff(y(x),x),x) = diff(y(x),x)^2*(1+diff(y(x),x)^2), y(x),'implicit')

Maple raw output

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