(Differential equation) Finding an exponential equation

[namegoeshere]

Homework Statement

A 44 gallon barrel of oil develops a leak at the bottom. Let $A(t)$ be the amount of oil in the barrel at a given time $t$. Suppose that the amount of oil is decreasing at a rate proportional to the product of the time elapsed and the amount of oil present in the barrel.

a. Give the mathematical model for A
b. Find the general solution of the differential equation
c. Find the particular solution for the initial condition

Homework Equations

The Attempt at a Solution

a.

$\frac{\partial{A}}{\partial{t}} = -Akt$​

b.

$\int{\frac{1}{A}\,dA} = \int{-kt\,dt}$
$\ln{A}=\frac{kt^2}{2}+C$
$A(t)=Ce^\frac{-kt^2}{2}$​

c.

$44=Ce^0$
$A(t)=44e^\frac{-kt^2}{2}$​

Is this correct? I'm not sure if I came up with the right $\frac{\partial{A}}{\partial{t}}$.

 

[haruspex]

All looks right to me.