First order linear differential equations and an application
If anyone can help me integrate the following first order differential equation, I'd appreciate it!...
dy=(y(tan x) + 2e^x)dx
I got it this far....
y = (C/abs(cos x))*(integral of e^x*abs(cos x)*dx)
And here's another problem....
An object falling near the earth's surface encounters air resistance that is proportional to its velocity. The acceleration due to gravity is -9.8m/s^2. So, without air resistance the object's acceleration can be modeled by the differential equation. dv/dt = -9.8. But aerodynamic drag represents a considerable retarding force as velocity increases. Thus a better model for an object falling near the surface of the earth is: dv/dt = kv - 9.8, where k is a constant of proportionality.
1. What are the units of k? Is k positive or negative?
2. Find the velocity of the object as a function of time if the initial velocity is V sub 0.
3. Use #2 to find the limit of the velocity as t -> infinity.
4. Find the position function s(t) of the object.