Therefore, the equation is = + C, which can be rewritten as y = 2x^2.
- y = sin−1(− cos x)
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. We can do
this easily by cross-multiplying. We get cos y dy = sin x dx. Next, we integrate both sides.
∫ cos y dy = ∫ sin x dx
sin y = − cos x + C
Now we solve for C. We plug in x = 0 and y = .
sin = − cos 0 + C
−1 = −1 + C
C = 0
Now we isolate y to get the equation y = sin−1(− cosx).
- 20,000 (approximately)
The phrase “grows exponentially” means that we can represent the situation with the
differential equation = ky, where k is a constant and y is the population at a time t. Here we
are also told that y = 4,000 at t = 0 and y = 6,500 at t = 3. 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 t variables on the other side. We can do this easily by dividing both sides by y
and multiplying both sides by dt. We get = k dt. Next, we integrate both sides.
∫ = ∫^ k dt
ln y = kt + C 0
