Question

Find the differential equation representing the family of curves \(y=2mx\).

Hint:

The number of arbitrary constants in the equation of a family of curves determines the order of the resulting differential equation. One constant means a first-order differential equation, two constants mean a second-order, and so on.

Answer 1

Step 1: Understanding the Concept:
To find the differential equation for a family of curves, we need to eliminate the arbitrary constant(s) from the given equation. Since there is one arbitrary constant ('m') in this equation, we will need to differentiate the equation once to get a second equation, and then use the two equations to eliminate 'm'.
Step 2: Key Formula or Approach:
1. Differentiate the given equation with respect to x. 2. From the original equation, express the constant in terms of x and y. 3. Substitute this expression for the constant into the differentiated equation.
Step 3: Detailed Explanation:
The given equation for the family of curves is: \[ y = 2mx \quad \text{---(1)} \] From equation (1), we can express the term with the constant 'm' as: \[ 2m = \frac{y}{x} \quad \text{---(2)} \] Now, differentiate the original equation (1) with respect to x: \[ \frac{d}{dx}(y) = \frac{d}{dx}(2mx) \] \[ \frac{dy}{dx} = 2m \quad \text{---(3)} \] Now we have an expression for \(2m\) in terms of the derivative, and another in terms of x and y. We can eliminate \(2m\) by substituting equation (2) into equation (3): \[ \frac{dy}{dx} = \frac{y}{x} \] To write this in a standard form, we can multiply both sides by x: \[ x \frac{dy}{dx} = y \] \[ x \frac{dy}{dx} - y = 0 \] Step 4: Final Answer:
The required differential equation is \(x \frac{dy}{dx} - y = 0\).