4.16.9 $y^{\prime }(x{)}^2=y(x{)}^2+1$

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

[_quadrature]

Book solution method
Missing Variables ODE, Independent variable missing, Solve for $y^{\prime }$

Mathematica ✓
cpu = 0.0182653 (sec), leaf count = 23

$$\{\{y(x)\to -\sinh (x-c_1)\},\{y(x)\to \sinh (c_1+x)\}\}$$

Maple ✓
cpu = 0.037 (sec), leaf count = 31

$$\{1+{\left(y\left(x\right)\right)}^2=0,x-\mathit{A}\mathit{r}\mathit{c}\mathit{s}\mathit{i}\mathit{n}\mathit{h}\left(y\left(x\right)\right)-\mathit{\_}\mathit{C}\mathit{1}=0,x+\mathit{A}\mathit{r}\mathit{c}\mathit{s}\mathit{i}\mathit{n}\mathit{h}\left(y\left(x\right)\right)-\mathit{\_}\mathit{C}\mathit{1}=0\}$$ Mathematica raw input

DSolve[y'[x]^2 == 1 + y[x]^2,y[x],x]

Mathematica raw output

{{y[x] -> -Sinh[x - C[1]]}, {y[x] -> Sinh[x + C[1]]}}

Maple raw input

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

Maple raw output

1+y(x)^2 = 0, x-arcsinh(y(x))-_C1 = 0, x+arcsinh(y(x))-_C1 = 0