Question

The slope at any point on the graph of the function h(x) where h(1)=9 is given by the expression dy/dx=
`2x√(y)`.

Write an expression for h(x) in terms of x.

Answer 1

The given differential equation is separable:

`(\mathrm dy)/(\mathrm dx) = 2x\sqrt y\implies(\mathrm dy)/(\sqrt y)=2x\,\mathrm dx`

Integrate both sides:

`2\sqrt y=x^2+C`

In this case,
`y=h(x)`, so that given
`h(1)=9`, we get

`2\sqrt9=1^2+C\implies C=5`

so that

`2√(h(x))=x^2+5\implies h(x)=\left(\frac{x^2+5}2\right)^2=\frac{x^4+10x^2+25}4`