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.