The order of the differential equation of the family of circles touching the y - axis at the origin is k, then the maximum value of `y=k cos x AA x in R` is

Answer

To solve the given problem step by step, we will follow the reasoning laid out in the video transcript.

### Step 1: Understand the Problem
We need to find the order of the differential equation of the family of circles that touch the y-axis at the origin.

### Step 2: Determine the Equation of the Circle
A circle that touches the y-axis at the origin has its center on the x-axis. Let’s denote the center of the circle as (a, 0). The radius of the circle is equal to the distance from the center to the y-axis, which is 'a'.

The general equation of the circle can be written as:
\[ (x - a)^2 + (y - 0)^2 = a^2 \]
Expanding this gives:
\[ x^2 - 2ax + a^2 + y^2 = a^2 \]
This simplifies to:
\[ x^2 + y^2 - 2ax = 0 \]

### Step 3: Differentiate the Equation
Now, we differentiate the equation with respect to x:
\[ \frac{d}{dx}(x^2 + y^2 - 2ax) = 0 \]
Using implicit differentiation:
\[ 2x + 2y \frac{dy}{dx} - 2a = 0 \]
Rearranging gives:
\[ 2y \frac{dy}{dx} = 2a - 2x \]
Thus, we can express 'a' as:
\[ a = x + y \frac{dy}{dx} \]

### Step 4: Substitute 'a' Back into the Circle Equation
Substituting the value of 'a' back into the circle equation:
\[ x^2 + y^2 - 2a x = 0 \]
Substituting for 'a':
\[ x^2 + y^2 - 2(x + y \frac{dy}{dx})x = 0 \]
This simplifies to:
\[ x^2 + y^2 - 2x^2 - 2xy \frac{dy}{dx} = 0 \]
Rearranging gives the differential equation:
\[ y^2 - x^2 = 2xy \frac{dy}{dx} \]

### Step 5: Determine the Order of the Differential Equation
The highest derivative in the equation is \(\frac{dy}{dx}\), which is of order 1. Thus, the order \(k\) of the differential equation is:
\[ k = 1 \]

### Step 6: Find the Maximum Value of \(y = k \cos x\)
Now, we need to find the maximum value of \(y = k \cos x\) where \(k = 1\):
\[ y = 1 \cos x = \cos x \]
The maximum value of \(\cos x\) is 1.

### Final Answer
Thus, the maximum value of \(y\) is:
\[ \boxed{1} \]