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?

A tank has 80 gallons of water and 20 kg of salt. Salty water with 2 kg/gal enters at 10 gal/min; the well-mixed solution leaves at 10 gal/min (volume stays constant at 80 gal). Let x(t) be the amount of salt (in kg) at time t. Write the ODE for x(t).

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

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