13: Intro to Differential Equations

The scent of a certain air freshener evaporates at a rate proportional to the amount of the air freshener present. Half of the air freshener evaporates within $2$ hours of being sprayed. If the scent of the air freshener is undetectable once $80\%$ has evaporated, how long will the scent of the air freshener last?

Find the particular solution that satisfies the given initial condition

$\left(y^2+y\right)e^{x}y^{\prime}=y^3+e^{x}y^3;y\left(0\right)=1$

Find the general solution to the differential equation.

$\frac{dy}{dx}=y\sqrt{x}$

$\$2,000$ is invested in an account that earns interest at a rate of $8.5\%$ and is compounded continuously. Find the particular solution that describes the growth of this account in dollars $A$ after $t$ years. Hint: When interest is compounded continuously, it grows exponentially with a growth constant equivalent to the interest rate.