The given differential equation is separable:
[tex]\dfrac{\mathrm dy}{\mathrm dx} = 2x\sqrt y\implies\dfrac{\mathrm dy}{\sqrt y}=2x\,\mathrm dx[/tex]
Integrate both sides:
[tex]2\sqrt y=x^2+C[/tex]
In this case, [tex]y=h(x)[/tex], so that given [tex]h(1)=9[/tex], we get
[tex]2\sqrt9=1^2+C\implies C=5[/tex]
so that
[tex]2\sqrt{h(x)}=x^2+5\implies h(x)=\left(\dfrac{x^2+5}2\right)^2=\dfrac{x^4+10x^2+25}4[/tex]
