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Suppose for a population P(t): the birth rate per person is proportional to P with proportionality constant 4; the death rate per person is proportional to \(P^2\) with proportionality constant 2. Write the ODE for \(dP/dt\).

A population has P(0)=8 and \(P(t)=\frac{{40}}{{8-3e^{{5t}}}}\). What happens to P(t) as t increases?

A student finds the general solution to \(\frac{dy}{dt}=5y\) and gives \(y=2e^{{5t}}\). What is the issue?

Consider \(y’=(y-3)^2\). After separation of variables we get the solutions: $$y=3-\frac{{1}}{{x+C}}$$ Find the solution satisfying \(y(0)=3\).

A tank has 200 gallons of water with 50 kg of salt. Pure water enters at 4 gal/min; the well-mixed solution leaves at 4 gal/min (volume stays constant at 200 gal). Let x(t) be the amount of salt (in kg) at time t. Write the ODE for x(t).

What is the integrating factor \(\rho(t)=e^{{\int P(t)\,dt}}\) for: $$\frac{dy}{dt}=30-5t^4y$$

What is the integrating factor \(\rho(x)=e^{{\int P(x)\,dx}}\) for: $$xy’+2y=\sin(x)$$

Find the general explicit solution for y: $$\frac{dy}{dx}=y^2+1$$

A population has P(0)=2 and \(P(t)=\frac{{12}}{{2+4e^{{-12t}}}}\). Compute \(\lim_{{t\to\infty}}P(t)\) and interpret this means.

An autonomous ODE has equilibria at \(P=5\) (unstable) and \(P=10\) (stable). If \(P(0)=3\), what does P(t) tend to as \(t\to\infty\)?