ODE satisfied by a special function

Context

I would like to estimate the distribution of the difference of two inverse gaussian variables. The convolution doesn't lead to any special functions according to Mathematica. Then, I would like to estimate it by another way, the ODE one.

Problem

I would like to derive an ode satisfied by a function defined as an integral. For $\lambda_1, \lambda_2, \mu_1, \mu_2$ four real positive numbers, I define two functions $f_1$ and $f_2$ as:

$$\forall x>0,~f_i(x) := x^{-3/2}\exp\left[-\frac{\lambda_i(x-\mu_i)^2}{2\mu_i^2x}\right].$$

Now, I define the the function $F$ as:

$$\forall z>0,~F(z) := \int_0^\infty f_1(x+z)f_2(x)dx.$$

The value $F(0)$ can be derived using a bit of algebras and modified Bessel functions of second kind. My question is the following, can we find an ode satisfied by $F$ on $\mathbb{R}_+$ ?

Simplification

Since $f_2$ is positive, the problem is equivalent to find an ode satisfied by $\varphi(z) := f_1(x+z)$.

I have tried to derive up to the second order and make some combinations of $\varphi$, $\varphi'$ and $\varphi''$ but doesn't give any ODE until now...

Any help will be highly appreciated!

Thank you!