Solving this DE containing an integral

asked 2015-07-19 07:13:44 -0600

FooBar gravatar image

updated 2015-07-20 15:37:31 -0600

calc314 gravatar image

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?

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