Linear equations and the integrating factor

Solve $y'+3 x^2 y = x^2$.

Solve $y'+ 2\sin(2x) y = 2\sin(2x)$, $y(\nicefrac{\pi}{2}) = 3$.

Suppose a water tank is being pumped out at 3 L/min. The water tank starts at 10 L of clean water. Water with a toxic substance is flowing into the tank at 2 L/min, with concentration $20t$ g/L at time $t$. When the tank is half empty, how many grams of toxic substance are in the tank (assuming perfect mixing)?

There are bacteria on a plate and a toxic substance is being added that slows down the rate of growth of the bacteria. That is, suppose that $\frac{dP}{dt} = (2-0.1\,t)P$. If $P(0) = 1000$, find the population at $t=5$.

A cylindrical water tank has water flowing in at $I$ cubic meters per second. Let $A$ be the area of the cross section of the tank in square meters. Suppose water is flowing out from the bottom of the tank at a rate proportional to the height of the water level. Set up the differential equation for $h$, the height of the water, introducing and naming constants that you need. You should also give the u…