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: $$xy’+2y=\sin(x)$$

A tank starts with 100 gallons of water. Water flows in at 3 gal/min and out at 5 gal/min. What is V(t)?Here V(t) denotes the volume (amount of gallons) of water in the tank of a function of time t in minutes.

A student finds the general solution to \(y”+ y=0\) and gives \(y=e^t\). What is the issue?

A population has P(0)=9 and \(P(t)=\frac{{54}}{{9-3e^{{12t}}}}\). What happens to P(t) as t increases?

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

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

What is the integrating factor \(\rho(x)=e^{{\int P(x)\,dx}}\) for: $$xy’+4y=30$$