Differentiate the given solution:

Substitute
and
into the ODE:

and it's easy to see the left side indeed reduces to 0.
-3 / 4 + p = 1/2
LCD 4 and 2 = 4
Multiply by LCD = 4
p. (4)- 3/4 .( 4 ) = 1/2.(4)
4 p - 3 = 2
Add 3 to both sides:
4 p - 3 + 3 = 2 + 3
4 p = 5
Divide both sides by 4 :
4p / 4 = 5 /4
p = 5/4
Okay, so we know it is going to be more than 32000, right?