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 problem, we need to find the order of the differential equation of the family of circles that touch the y-axis at the origin and then determine the maximum value of \( y = k \cos x \).

### Step 1: Understanding the Circle's Equation
The family of circles that touch the y-axis at the origin can be represented by their center lying on the x-axis. Let's denote the center of the circle as \( (a, 0) \) where \( a \) is the radius of the circle. The equation of the circle can be written as:
\[ (x - a)^2 + y^2 = a^2 \]

### Step 2: Differentiate the Circle's Equation
Next, we differentiate the equation of the circle with respect to \( x \):
\[ 2(x - a) + 2y \frac{dy}{dx} = 0 \]
This simplifies to:
\[ (x - a) + y \frac{dy}{dx} = 0 \]

### Step 3: Express \( a \) in Terms of \( x \) and \( y \)
From the differentiated equation, we can express \( a \):
\[ a = x - y \frac{dy}{dx} \]

### Step 4: Substitute \( a \) Back into the Circle's Equation
Now, we substitute \( a \) back into the original circle equation:
\[ (x - (x - y \frac{dy}{dx}))^2 + y^2 = (x - y \frac{dy}{dx})^2 \]
This simplifies to:
\[ (y \frac{dy}{dx})^2 + y^2 = (x - y \frac{dy}{dx})^2 \]

### Step 5: Rearranging the Equation
Expanding and rearranging gives us a differential equation. The highest derivative present will determine the order of the differential equation.

### Step 6: Determine the Order of the Differential Equation
The highest derivative in our equation is \( \frac{dy}{dx} \), which is of first order, thus the order of the differential equation is:
\[ k = 1 \]

### Step 7: Find the Maximum Value of \( y = k \cos x \)
Given \( k = 1 \), we have:
\[ y = \cos x \]
The maximum value of \( \cos x \) is well-known:
\[ \text{Maximum value of } \cos x = 1 \]

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