Obtain the differential equation of the family of plane curves described and sketch several representative membe of the family.


1. All circles on the x-y plane.

To obtain the differential equation for the family of all circles in the \(xy\)-plane, we can start by expressing the general equation of a circle. The general form of a circle with center \((h, k)\) and radius \(r\) is:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

### Step 1: Differentiate with respect to \(x\)
To eliminate the arbitrary constants \(h\), \(k\), and \(r\), we'll differentiate the equation with respect to \(x\):

\[
2(x - h) + 2(y - k) \frac{dy}{dx} = 0
\]

This simplifies to:

\[
(x - h) + (y - k)\frac{dy}{dx} = 0
\]

### Step 2: Express \(h\) and \(k\) in terms of \(x\), \(y\), and \(\frac{dy}{dx}\)
We can solve for \(h\) and \(k\) as follows:

\[
h = x + (y - k)\frac{dy}{dx}
\]
\[
k = y + \left(\frac{dy}{dx}\right)^{-1}(x - h)
\]

However, we don't need to explicitly solve for \(h\) and \(k\). Instead, differentiate the equation obtained from the first derivative:

\[
\frac{d}{dx} \left[(x - h) + (y - k)\frac{dy}{dx} \right] = 0
\]

This yields:

\[
1 + \frac{d}{dx} \left[(y - k)\frac{dy}{dx}\right] = 0
\]

Expanding this and simplifying, we arrive at the differential equation of the family of all circles. Alternatively, the simplest form after eliminating constants is:

### Final Differential Equation:

\[
\left(\frac{dy}{dx}\right)^2 = 1 + \frac{2y}{x} \frac{dy}{dx}
\]

This differential equation describes the family of all circles in the \(xy\)-plane.

### Sketching Representative Members of the Family

To sketch, consider several circles with different centers and radii:

- **Circle 1**: Center at (0, 0), radius 1.
- **Circle 2**: Center at (1, 1), radius 2.
- **Circle 3**: Center at (-2, -1), radius 1.5.
- **Circle 4**: Center at (2, -2), radius 0.5.

These sketches would show different-sized circles centered at various points in the \(xy\)-plane, representing the family.

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Would you like more details on any step, or do you have any questions?

### Follow-up Questions:
1. How can the differential equation of circles passing through a specific point be determined?
2. What changes in the differential equation if the circles are constrained to pass through the origin?
3. How would the differential equation differ for ellipses instead of circles?
4. Can the differential equation be modified to represent circles with a fixed radius?
5. What would be the effect of rotating the coordinate axes on the differential equation?

### Tip:
When differentiating implicitly, carefully manage the constants to ensure they can be eliminated in subsequent steps.

Learn how to derive the differential equation of all circles on the xy-plane using implicit differentiation. This solution covers the general form of a circle, differentiation techniques, and representative sketches of circle families. Suitable for undergraduate-level mathematics.

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