What is the order of the differential equation of all ellipses whose axes are along the coordinate axes?
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Finding the Order of Differential Equation for Ellipses with Axes Along Coordinate Axes
The question asks for the order of the differential equation that represents all ellipses whose axes are aligned with the coordinate axes. To find the order of a differential equation representing a family of curves, we first write the general equation of the family of curves and then determine the number of arbitrary constants present in that equation. The order of the resulting differential equation will be equal to the number of independent arbitrary constants we need to eliminate through differentiation.
Equation of Ellipses with Axes Along Coordinate Axes
The standard equation for an ellipse centered at the origin (whose axes are along the coordinate axes) is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Here, \(x\) and \(y\) are variables, and \(a\) and \(b\) are constants representing the lengths of the semi-major and semi-minor axes. For a family of all such ellipses, \(a\) and \(b\) are arbitrary constants that can vary from one ellipse to another.
Identifying Arbitrary Constants
In the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), the constants \(a^2\) and \(b^2\) are the arbitrary parameters defining the specific ellipse. There are two independent arbitrary constants, \(a\) and \(b\) (or \(a^2\) and \(b^2\)). For example, we can choose \(a=3, b=2\) for one ellipse, and \(a=4, b=1\) for another. Since there are two such independent parameters, we expect the differential equation representing this family of ellipses to be of order 2.
Arbitrary Constants and Differential Equation Order
A fundamental concept in forming differential equations from families of curves is that the order of the differential equation is equal to the number of essential arbitrary constants in the equation of the family of curves. Since the equation of ellipses with axes along the coordinate axes, \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), contains two arbitrary constants (\(a\) and \(b\)), the order of the differential equation representing this family will be 2.
Deriving the Differential Equation (Optional Step for Clarity)
Let's briefly show how the differential equation is formed to confirm the order. Starting with:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Differentiate with respect to \(x\):
\[ \frac{2x}{a^2} + \frac{2y}{b^2} \frac{dy}{dx} = 0 \]
Divide by 2 and let \(y' = \frac{dy}{dx}\):
\[ \frac{x}{a^2} + \frac{y}{b^2} y' = 0 \]
Rearrange to isolate the ratio of constants:
\[ \frac{x}{a^2} = - \frac{y}{b^2} y' \implies \frac{b^2}{a^2} = - \frac{y y'}{x} \]
This equation still contains the arbitrary constants (as a ratio). Differentiate this result or the previous differentiated equation \( \frac{x}{a^2} + \frac{y}{b^2} y' = 0 \) again with respect to \(x\):
Differentiating \( \frac{x}{a^2} + \frac{y}{b^2} y' = 0 \):
\[ \frac{1}{a^2} (1) + \frac{1}{b^2} (y' \cdot y' + y \cdot y'') = 0 \]
\[ \frac{1}{a^2} + \frac{1}{b^2} ((y')^2 + y y'') = 0 \]
Multiply by \(a^2 b^2\):
\[ b^2 + a^2 ((y')^2 + y y'') = 0 \]
From \( \frac{b^2}{a^2} = - \frac{y y'}{x} \), substitute \(b^2 = -a^2 \frac{y y'}{x}\) into the equation above:
\[ -a^2 \frac{y y'}{x} + a^2 ((y')^2 + y y'') = 0 \]
Since \(a \neq 0\), we can divide by \(a^2\):
\[ - \frac{y y'}{x} + (y')^2 + y y'' = 0 \]
Multiply by \(x\):
\[ -y y' + x (y')^2 + x y y'' = 0 \]
Rearrange:
\[ x y y'' + x (y')^2 - y y' = 0 \]
This is the differential equation for the family of ellipses with axes along the coordinate axes.
Determining the Order from the Derived Equation
The derived differential equation is \( x y y'' + x (y')^2 - y y' = 0 \). The highest order derivative present in this equation is \(y''\), which is the second derivative of \(y\) with respect to \(x\).
The order of a differential equation is the order of the highest derivative occurring in the equation.
Conclusion on the Order
Since the highest derivative in the differential equation \( x y y'' + x (y')^2 - y y' = 0 \) is the second derivative (\(y''\) or \( \frac{d^2y}{dx^2} \)), the order of the differential equation is 2.
This matches the number of arbitrary constants we identified initially.
| Concept | Description | Value/Outcome |
|---|---|---|
| Family of Curves | Ellipses with axes along coordinate axes | \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) |
| Arbitrary Constants | Parameters defining specific curves in the family | \(a\) and \(b\) (or \(a^2\) and \(b^2\)) |
| Number of Arbitrary Constants | Count of independent constants | 2 |
| Order of Differential Equation | Highest derivative order needed to eliminate constants | Equals the number of arbitrary constants |
| Derived Differential Equation | Equation free of arbitrary constants | \( x y y'' + x (y')^2 - y y' = 0 \) |
| Highest Derivative | Highest order of differentiation in the equation | \(y''\) (second derivative) |
| Final Order | Order of the differential equation | 2 |
Therefore, the order of the differential equation of all ellipses whose axes are along the coordinate axes is 2.
Revision Table: Differential Equation Order
Let's quickly review the key concepts related to the order of a differential equation:
- Differential Equation: An equation involving an unknown function and its derivatives.
- Order of a Differential Equation: The order of the highest derivative present in the equation.
- Family of Curves: A set of curves described by an equation containing one or more arbitrary constants (parameters).
- Forming a Differential Equation: The process of eliminating arbitrary constants from the equation of a family of curves by repeated differentiation.
- Relationship: The order of the differential equation formed from a family of curves is equal to the number of independent arbitrary constants in the family's equation.
Additional Information: Forming Differential Equations
When forming the differential equation for a family of curves, the goal is to obtain an equation that holds true for every member of the family, regardless of the specific values of the arbitrary constants. This is achieved by differentiating the equation of the family repeatedly and algebraically manipulating the resulting equations to eliminate the constants. If there are \(n\) independent arbitrary constants, you will generally need to differentiate \(n\) times to eliminate them, resulting in a differential equation of order \(n\). Examples:
- Family of straight lines \(y = mx + c\): Two arbitrary constants \(m\) and \(c\). Requires two differentiations. \(y' = m\), \(y'' = 0\). Order is 2.
- Family of circles with center at origin \(x^2 + y^2 = r^2\): One arbitrary constant \(r\) (or \(r^2\)). Requires one differentiation. \(2x + 2y y' = 0 \implies x + y y' = 0\). Order is 1.
In our case, for ellipses \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), we have two arbitrary constants \(a\) and \(b\), leading to a second-order differential equation.
Similar Questions
Consider the following in respect of the differential equation:
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Which of the above statements is/are correct?
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Which of the statements given above is/are correct?
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Important Questions from Order and Degree of a Differential Equation
Consider the following in respect of the differential equation:
\(\frac{{{d^2}y}}{{d{x^2}}} + 2{\left( {\frac{{dy}}{{dx}}} \right)^2} + 9y = x\)
1. The degree of the differential equation is 1.
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Which of the above statements is/are correct?
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