- y =
We solve this differential equation by separation of variables. We want to get all of the y
variables on one side of the equals sign and all of the x variables on the other side. First, we
factor the y out of the denominator of the right hand expression: = = .
Next, we multiply both sides by y and by dx. We get y dy = . Next, we integrate both
sides.
= tan−1 x + C
Now we isolate y: y^2 = 2tan−1 x + C.
Now we solve for C. We plug in x = 0 and y = 2.
C = 4
Therefore, the equation is y = .
- ≈ 1,800 ft^3
We can express this situation with the differential equation = kV, where k is a constant and
V is the volume of the sphere at time t. We are also told that V = 36π when t = 0 and V = 90π
when t = 1. We solve this differential equation by separation of variables. We want to get all of
the V variables on one side of the equals sign and all of the t variables on the other side. We
can do this easily by dividing both sides by V and multiplying both sides by dt. We get = k
