# Solving this DE containing an integral

I am trying to solve

$$0 = - \partial_a F(a)-e(a)F(a) + \int_0^{a} e(a')\partial_a F(a') d a' + n$$

optimally, without specifying e(a). But if necessary (as I guess), e(a) = k1exp(k2a). Here's my code:

var('a b k_1 k_2 n')
e(b) = k_1*exp(b*k_2)
F  = function('F', a)
g(b) = e(b)*diff(F,b,1)
de = -diff(F, a, 1) - e(a)*F(a) + g(b).integral(b, 0, a) + n
y = desolve(de, F, ivar=a); y


And the output is

(_C + n*Ei(k_1*e^(a*k_2)/k_2)/k_2)*e^(-k_1*e^(a*k_2)/k_2)


However, I believe that something is wrong with my integral, it is not doing what I expect it to do. For example, if I change the integration boundary to 1:

    de = -diff(F, a, 1) - e(a)*F(a) + g(b).integral(b, 0, a) + n


I will still get the same result:

(_C + n*Ei(k_1*e^(a*k_2)/k_2)/k_2)*e^(-k_1*e^(a*k_2)/k_2)


Where am I going wrong?

edit retag close merge delete