The general solution of a given differential equation contains arbitrary constants depending on the order of the differential equation. Let's understand this in detail:

1. Differential Equation:
A differential equation is an equation that relates a function with its derivatives. It represents a relationship between the function and its rate of change.

2. Order of a Differential Equation:
The order of a differential equation is the highest order of the derivative present in the equation. For example, if the equation involves only the first derivative of the function, it is a first-order differential equation. Similarly, if it involves the second derivative, it is a second-order differential equation, and so on.

3. Particular Solution:
A particular solution is a specific solution of a differential equation that satisfies the given initial conditions.

4. General Solution:
The general solution of a differential equation is a family of solutions that contains all possible solutions of the equation. It includes a set of functions that satisfy the differential equation, along with arbitrary constants.

5. Arbitrary Constants:
Arbitrary constants are constants that can take any value. They are introduced in the general solution to account for the different solutions that satisfy the given differential equation.

6. Number of Arbitrary Constants:
The number of arbitrary constants in the general solution depends on the order of the differential equation.

- For a first-order differential equation, the general solution contains exactly one arbitrary constant.
- For a second-order differential equation, the general solution contains exactly two arbitrary constants.
- For higher-order differential equations, the general solution contains arbitrary constants equal to the order of the differential equation.

7. Importance of Arbitrary Constants:
The arbitrary constants in the general solution allow us to find the particular solution by substituting specific values for the constants. These values are determined by the given initial conditions or boundary conditions.

In conclusion, the general solution of a given differential equation contains arbitrary constants depending on the order of the differential equation. These arbitrary constants allow us to find particular solutions that satisfy the given initial or boundary conditions.