Question

The differential equation of the family of circlen with centre as origin is,

Short answer

The differential equation is \( \frac{dy}{dx} = -\frac{x}{y} \).

Step-by-step solution

Understand the Equation of a Circle

The general equation of a circle with its center at the origin (0, 0) and radius \( r \) is given by:\[ x^2 + y^2 = r^2 \]Since \( r \) can be any positive constant, this represents a family of circles centered at the origin.

Differentiate the Equation

To find the differential equation, differentiate the circle's equation with respect to \( x \). The differentiation of \( x^2 + y^2 = r^2 \) with respect to \( x \) gives:\[ 2x + 2y \frac{dy}{dx} = 0 \]

Solve for the Differential Equation

Rearrange the differentiated equation to solve for \( \frac{dy}{dx} \):\[ 2y \frac{dy}{dx} = -2x \]Divide both sides by \( 2y \) to isolate \( \frac{dy}{dx} \):\[ \frac{dy}{dx} = -\frac{x}{y} \]

Finalize the Differential Equation

The differential equation \( \frac{dy}{dx} = -\frac{x}{y} \) represents the family of circles with centers at the origin. This means every solution \( y(x) \) of this differential equation will represent a part or whole of a circle centered at the origin.

Key concepts

Equation of a Circle

The equation of a circle is a fundamental concept in geometry. A circle is a set of all points in a plane that are equidistant from a particular point called the center. In this case, the center is at the origin of the coordinate system, that is, the point (0, 0). For such a circle, if the radius is denoted as \( r \), the equation simplifies to \( x^2 + y^2 = r^2 \). This equation captures all points \((x, y)\) that lie at a distance \( r \) from the origin.

• The term \( x^2 + y^2 \) represents the sum of the squares of the distances from a point on the circle to the x-axis and y-axis, respectively.
• The equation states that this sum is constant and equal to \( r^2 \).

Understanding this concept provides a basis for dealing with more intricate problems involving circles.

Differentiation

Differentiation is a fundamental tool in calculus. It's the process of finding a derivative, which measures how a function changes as its input changes. In the context of the equation \( x^2 + y^2 = r^2 \), differentiation with respect to \( x \) helps us find the slope of the curve at any point. This is especially useful for identifying the behavior of circles and other geometric shapes.
When differentiating \( x^2 + y^2 = r^2 \), treat \( y \) as a function of \( x \). This approach not only uses the product and chain rules of differentiation but also reflects the concept of implicit differentiation, which is used here when we differentiate \( y^2 \) as \( 2y \cdot \frac{dy}{dx} \).

• The derivative \( \frac{dy}{dx} \) gives the rate of change of \( y \) with respect to \( x \).
• This represents the circle's instantaneous slope or tangent at any point on its circumference.

Origin-Centered Circles

Origin-centered circles are those whose centers lie at the point (0, 0) of a coordinate system. Understanding them is important because they simplify many algebraic operations and offer symmetric properties that are easy to manipulate mathematically.

• Their equation, \( x^2 + y^2 = r^2 \), indicates that all points on the circle are equidistant (radius \( r \)) from the origin.
• Such circles exhibit radial symmetry; any line through the origin will intersect the circle in two symmetrical points.

This property is not only useful in solving problems but also in visualizing transformations and other interactions involving circles. These symmetrical properties make calculations involving origin-centered circles particularly straightforward compared to those with non-origin centers.

Family of Curves

A family of curves is a set of curves that are related by an altering parameter, such as radius in the case of circles centered at the origin. The general equation \( x^2 + y^2 = r^2 \) does not specify a particular circle, but rather, a whole family of circles with varying radii.
A differential equation generalizes this family by compelling how \( y \) changes with respect to \( x \). In this scenario, \( \frac{dy}{dx} = -\frac{x}{y} \) describes all possible circles that could be drawn with the origin as their center.

• Each value of \( r \) corresponds to a distinct circle within this family.
• The differential equation \( \frac{dy}{dx} = -\frac{x}{y} \) encompasses infinitely many curves because there are infinitely many values for \( r \).

The concept of a family of curves is practical in many scientific fields where understanding variations within a similar group of curves is essential.