Number of Arbitrary Constants in the Complete Primitive of Differential Equation:

To determine the number of arbitrary constants in the complete primitive of the given differential equation, we first need to find the general solution of the equation.

Step 1: Rewrite the Differential Equation
The given differential equation can be rewritten as:

(d^5y)/(dx^5) + 2 * (d^4y)/(dx^4) = 0

Step 2: Find the Characteristic Equation
To find the general solution, we need to solve the characteristic equation associated with the given differential equation. The characteristic equation is obtained by assuming the solution to be of the form y = e^(mx), where m is a constant.

The characteristic equation for the given differential equation is:

m^5 + 2m^4 = 0

Step 3: Solve the Characteristic Equation
Solving the characteristic equation, we get:

m^4(m + 2) = 0

This equation has two solutions: m = 0 and m = -2.

Step 4: General Solution
The general solution of the differential equation can be expressed as a linear combination of the solutions obtained from the characteristic equation:

y = C1 + C2 * x + C3 * x^2 + C4 * x^3 + C5 * x^4

where C1, C2, C3, C4, and C5 are arbitrary constants.

Step 5: Number of Arbitrary Constants
From the general solution, we can see that there are 5 arbitrary constants (C1, C2, C3, C4, and C5). Therefore, the number of arbitrary constants in the complete primitive of the given differential equation is 5.

Conclusion:
In the complete primitive of the given differential equation, there are 5 arbitrary constants. These constants represent the infinite family of solutions that satisfy the differential equation.