ODE Instability with sinh and cosh functions in Julia

Question

I want to solve the first-order differential equation $$ \begin{align} \frac{d\alpha}{d\phi} = \frac{\phi \sigma^2 \sin(2d\alpha)+2d\sinh(\sigma^2\alpha \phi)}{-\alpha \sigma^2 \sin(2d\alpha) +2d\cos(2d\alpha)+2d\cosh(\sigma^2\alpha \phi)} \;, \end{align}$$ where $d$ and $\sigma$ are fixed. I can't obtain a solution for this differential equation, and I guess the source of my problem lies in the sinh and cosh functions. Indeed, I can solve for example $d\alpha/d\phi=\phi \sigma^2 \sin(2d\alpha)$ with no trouble at all, but $d\alpha/d\phi=2d\sinh(\sigma^2\alpha \phi)$ returns Warning: Instability detected. Aborting. Why can I numerically solve a differential equation with sine but not with sinh? I put below a MWE reproducing the error.

using DifferentialEquations
using Plots
            
const global s2=81. # σ^2
const global d=-0.00090         

function ode(du,u,p,t)
    α = u[1]
    du[1] = 2*d*sinh(s2*α*t)
end
α0 =[1.0] # This initial value is just a guess
prob = ODEProblem(ode,α0,(-5.0,5.0))
sol = solve(prob)
plot(sol,label="2*d*sinh(s2*α*ϕ)")

Answer 1

I inserted a print statement into the RHS function:

function ode(du,u,p,t)
    α = u[1]
    @show d,s2,α,t,2*d*sinh(s2*α*t)
    du[1] = 2*d*sinh(s2*α*t)
end

Here's what it showed for the very first derivative calculation:

(d, s2, α, t, 2 * d * sinh(s2 * α * t)) = (-0.0009, 81.0, 1.0, -5.0, 6.974413306346224e172)

Let me repeat: your derivative is $6.97 \times 10^{172}$ at your starting $t=-5$. this is because $s2 * \alpha * t = -405.0$ and $\sinh(-405) \approx 6.97 \times 10^{172}$. When your derivative is that gigantic, your equation is diverging almost infinitely fast which explains the behavior.