An autonomous ODE has equilibria at \(P=5\) (unstable) and \(P=10\) (stable). If \(P(0)=7\), what does P(t) tend to as \(t\to\infty\)?

The Embargo Act of 1806  by Jefferson was most criticized by the Federalists

Solve \(\frac{dv}{dt}=\frac{4t}{2v}\) where \(v(0)=-3\)

The acceleration of a particle equals the negative of its position x(t). Write the ODE for x(t).

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

An autonomous ODE has equilibria at \(P=5\) (unstable) and \(P=10\) (stable). If \(P(0)=10\), what is P(t)?

Goal: solve \(y’+\frac{{2}}{{x}}y=x\). Integrating factor \(\rho=x^2\) gives: $$(x^2y)’=x^3$$ After integrating we get \(x^2y=\frac{{x^4}}{{4}}+C\). What is the correct solution for y?

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

The graph of y(x) has the property that the tangent line at any point (x, y) passes through the point (0, 3). Write an ODE for y(x).

A student finds the general solution to \(\frac{dy}{dx}=\frac{{1}}{{x+3}}\) and gives \(y=\ln|x+3|+C\). What is the issue?