Start by noting that the identically zero function for any of the three is invalid by the division part of the equations, so multiplying all of the equations by
is a valid operation.
Then we have:
Add these together and note that we are in a situation where we have applied the product rule to the function
, so we get:
The semi-clever replacement is to note that
So solving the differential equation:
is simple, it gives that
by use of the characteristic equation method.
Now the initial condition
tells us that
Now returning to the equation
we can see by simple separation of variables that
The second integral coming from adding and subtracting 1 and making a u-substitution.
Using our initial conditions, we find that
From there it's a simple exponentiation to recover