Question

Write and then solve the differential equation for the statement: "The rate of change of y with respect to x is inversely proportional to the square root of y."

Answer 1

Brief review of proportionality relationships:

When two quantities $a,b$ are $\textbf{directly}$ proportional, that means any change in $a$ manifests a $\textbf{direct}$ change (think "in the same direction") in $b$.

Silly example: "The more I eat, the fatter I get." Here the amount one eats is directly proportional to one's body weight.

This change isn't always one-for-one, so we introduce a constant $k$ to account for any scaling that occurs on either variables behalf. In general, though, we can write a directly proportional relationship as $a=kb$.

Now, when $a,b$ are $\textbf{inversely}$ proportional, then a change in $a$ manifests a change in $b$ in the $\textbf{inverse}$ (opposite) direction.

Silly example: "The more I eat, the less thin I get."

This time we write the relation as $ab=k$.

To get back to your problem: To say that the rate of change of $y(x)$ is inversely proportional to $\sqrt y$ is to say that there is some constant $k$ such that

$\sqrt y\dfrac{\mathrm dy}{\mathrm dx}=k$

This is a separable ODE:

$y^{1/2}\,\mathrm dy=k\,\mathrm dx$
$\displaystyle\int y^{1/2}\,\mathrm dy=\int k\,\mathrm dx$
$\dfrac23y^{3/2}+C_y=kx+C_x$
$\dfrac23y^{3/2}=kx+C$
$y^{3/2}=\dfrac{3k}2x+C$
$y=\left(\dfrac{3k}2x+C\right)^{2/3}$