Example 3

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.