Solution to Integral Equation obtained by modifying Newsvendor Model

Question

I have the following equation and want to find p(q)

$$ a p(q)+b-2 c \int_{0}^{q} p(x) d x=0 $$

I tried to differentiate, but do not know how to proceed with this integral equation. Please help. Here a,b and c are constants. p(0)=0.

Answer 1

We may differentiate the given equation

$ap(q) + b - 2c \displaystyle \int_0^q p(x) \; dx = 0 \tag 1$

with respect to $q$, and obtain

$ap'(q) - 2c p(q) = 0; \tag 2$

taking

$q = 0 \tag 3$

we obtain from (1)

$ap(0) + b = 0, \tag 4$

or

$p(0) = -\dfrac{b}{a}, \tag 5$

which we take as the initial condition for the ordinary differential equation (2); the well-known solution is then

$p(q) = p(0)e^{2cq/a} = -\dfrac{b}{a} e^{2cq/a}. \tag 6$

Note Added in Edit, Friday 16 October 2020 7:03 PM PST: First, a few remarks in response to the comments of our OP Conorlash, which may be found appended to this answer. Having followed the links he/she provided, I am forced to admit that my understanding of the economic/financial applications of this model is still severely limited, so I am afraid I cannot offer any useful interpretive insights at the present time. Meanwhile, we may continue to investigate (1) from a purely mathematical point of view. To this ende we note that if our OP's suggested initial condition

$p(0) = 0 \tag 7$

is adopted, then for any $q$ we have, in light of (6),

$p(q) = p(0)e^{2cq/a} = -0 \cdot \dfrac{b}{a} e^{2cq/a} = 0, \tag 8$

assuming of course

$a \ne 0, \tag 9$

as is evident from (5), which indicates that

$b = 0 \tag{10}$

is equivalent to (7) under the condition (9).

Finally, it should be observed that assigning different values to the coefficients $a$, $b$ and $c$ will have consequences in terms of the solution space; the engaged reader my discover these through further simple analyses of the cases of (1). End of Note.