Differential Equations
Example 3
Solve: dy + 2ydx = 6 dx given x = 0 when y = 1.
Separate variables:
dy = (6 – 2y) dx
dy/(6 – 2y) = dx
Integrate, solve for y, then C:
-ln (6 – 2y)/2 = x + C
ln (6 – 2y) = 2x + 2C
Note that 2C is still a constant. Let’s call it C’.
ln (6 – 2y) = 2x + C’
Now take the exponential of both sides:
6 – 2y = e-2x + C’
6 – 2y = e-2x * eC’
And eC’ is a constant. Let’s call it C”.
6 – 2y = e-2x * C”
Or: 6 – 2y = C”e-2x
This is a good place to solve for our constant, C”.
6 – 2(1) = C”e-2(0)
4 = C”
6 – 2y = 4e-2x
Solve for y: y = 3 – 2e2x
Whew! That was a long one. Study it over again. We’ll be seeing more like it later.
