Is the Solution to y'=\sqrt{y^2+x^2+1} Defined for All x and Greater Than sinhx?

[Malmstrom]

Consider the problem

$$y'=\sqrt{y^2+x^2+1}$$
$$y(0)=0$$

Prove that the solution is defined for all $$x \in \mathbb{R}$$ and that $$y(x) \geq \sinh (x)$$ $$\forall x \geq 0$$

 

[Redbelly98]

Consider the problem

$$y'=\sqrt{y^2+x^2+1}$$
$$y(0)=0$$

Prove that the solution is defined for all $$x \in \mathbb{R}$$ and that $$y(x) \geq \sinh (x)$$ $$\forall x \geq 0$$

Is this homework?

 

[Malmstrom]

Not really, I just don't know where to start from.