Number of arbitrary constant in the general solution of a differential equation is equal to order of D.E.

Answer

To understand the relationship between the number of arbitrary constants in the general solution of a differential equation and the order of that differential equation, we can break it down step by step.

### Step-by-Step Solution:

1. **Definition of Differential Equation**:
A differential equation is an equation that involves derivatives of a function. The order of a differential equation is defined as the highest derivative present in the equation.

**Hint**: Remember that the order indicates how many times a function is being differentiated.

2. **Understanding Arbitrary Constants**:
When we solve a differential equation, we often find that the solution includes arbitrary constants. These constants arise because the solution to a differential equation is not unique; it can represent a family of functions.

**Hint**: Arbitrary constants are placeholders for values that can vary, reflecting the generality of the solution.

3. **Example of a First-Order Differential Equation**:
Consider the first-order differential equation derived from the family of lines passing through the origin:
\[ y = mx \]
Here, \(m\) is an arbitrary constant representing the slope of the line.

**Hint**: The number of arbitrary constants corresponds to the degree of freedom in the solution.

4. **Differentiation**:
If we differentiate \(y = mx\) with respect to \(x\), we get:
\[ \frac{dy}{dx} = m \]
This shows that the order of the differential equation is 1, as we have one derivative.

**Hint**: The differentiation process helps us identify the order of the differential equation.

5. **Forming the Differential Equation**:
Rearranging the equation gives us:
\[ x \frac{dy}{dx} - y = 0 \]
This is a first-order differential equation, confirming that the order is 1.

**Hint**: Rearranging the equation can help clarify the relationship between variables and derivatives.

6. **General Solution and Constants**:
The general solution of this first-order differential equation will contain one arbitrary constant, consistent with the order of the differential equation.

**Hint**: The number of arbitrary constants in the general solution is equal to the order of the differential equation.

7. **Higher-Order Differential Equations**:
For a second-order differential equation, such as one derived from the equation of concentric circles:
\[ x^2 + y^2 = r^2 \]
Differentiating twice will yield a solution that includes two arbitrary constants.

**Hint**: As the order increases, so does the number of arbitrary constants in the general solution.

### Conclusion:
The number of arbitrary constants in the general solution of a differential equation is indeed equal to the order of the differential equation. This principle holds true across different types of differential equations, whether they are first-order, second-order, or higher.