Question

The differential equation of all conics having centre at the origin is of order. (A) 2 (B) 3 (C) 4 (D) 5

Short answer

The differential equation of all conics having center at the origin is of order 2. This is represented by the equation \((B+2Cy)\frac{\mathrm{d^2}y}{\mathrm{d}x^2} + (2A)(2) = 0\), where the highest derivative is the second derivative, indicating an order of 2. Therefore, the correct answer is (A) 2.

Step-by-step solution

Write the general equation of a conic with center at the origin

The general equation of a conic with its center at the origin can be represented as \(Ax^2 + Bxy + Cy^2 = 0\), where A, B, and C are constants. This equation represents the family of conics we want to find the order of the corresponding differential equation for.

Differentiate the general equation implicitly with respect to x

Differentiate the given general equation implicitly with respect to x: \[\frac{\mathrm{d}}{\mathrm{d}x} (Ax^2 + Bxy + Cy^2) = \frac{\mathrm{d}}{\mathrm{d}x} (0)\] Applying the derivatives, we get: \[2Ax + By +B\frac{\mathrm{d}y}{\mathrm{d}x}+ 2Cy\frac{\mathrm{d}y}{\mathrm{d}x} = 0\] Now, we can rewrite this equation as: \[\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{-2Ax - By}{B + 2Cy}\]

Differentiate the expression for dy/dx again with respect to x

Now, we have to differentiate the expression for \(\frac{\mathrm{d}y}{\mathrm{d}x}\) with respect to x again. Since it is given in the form of y, there will be terms containing \(\frac{\mathrm{d}y}{\mathrm{d}x}\) and will have \(\frac{\mathrm{d^2}y}{\mathrm{d}x^2}\) after simplification. Differentiating it with respect to x, we get: \[\frac{\mathrm{d^2}y}{\mathrm{d}x^2} = -\frac{(2A)(2) + -(2C)\frac{\mathrm{d^2}y}{\mathrm{d}x^2} + (B+2Cy)\frac{\mathrm{d^2}y}{\mathrm{d}x^2}}{(B+2Cy)^2}\] Taking the LCM and solving for \(\frac{\mathrm{d^2}y}{\mathrm{d}x^2}\): \[(B+2Cy)\frac{\mathrm{d^2}y}{\mathrm{d}x^2} + (2A)(2) = 0\]

Determine the order of the differential equation

Now, we can analyze the final equation we got. The highest order of the derivative present in the equation is 2: \[(B+2Cy)\frac{\mathrm{d^2}y}{\mathrm{d}x^2} + (2A)(2) = 0\] Since the highest derivative is the second derivative, the order of the differential equation is 2.

Match the result with the given options

Our result, order = 2, matches with option (A): (A) 2 So, the correct answer is (A) 2.

Key concepts

Conics with Center at Origin

A conic section is a curve obtained as the intersection of a cone with a plane. When these conics have their centers at the origin, they take a simpler form. The general equation for such conics can be written as:

• \(Ax^2 + Bxy + Cy^2 = 0\)
• Here, \(A\), \(B\), and \(C\) are constants.

This representation includes different types of conics such as circles, ellipses, parabolas, and hyperbolas, all centered at the origin (0,0). Understanding this general form helps lay the foundation for finding the differential equations that describe these shapes.

Order of Differential Equation

In mathematics, differential equations describe the relationship between a function and its derivatives. The order of a differential equation is determined by the highest derivative present in the equation. For the conics centered at the origin, we derive a differential equation starting from the general form. By analyzing this equation, we observe that the highest order of derivative is the second derivative. Therefore, the order of the differential equation is 2. Recognizing this order is important because it helps in solving and understanding the behavior of the system described by the equation.

Implicit Differentiation

Implicit differentiation is a technique used when a function is not given in an explicit form, like \(y = f(x)\). Instead, the variables are intermingled, such as in the equation of a conic \(Ax^2 + Bxy + Cy^2 = 0\). Here’s how implicit differentiation works:

• Differentiate every term with respect to \(x\), considering \(y\) as a function of \(x\).
• Apply the product rule when needed, especially for terms like \(Bxy\).

Using implicit differentiation helps us derive the relationship between \(x\) and \(y\) that can reveal valuable insights into the characteristics of the curve. It’s a crucial step in reaching the differential equation.

Second Derivative

The second derivative represents the rate of change of the rate of change. It provides insight into the concavity and acceleration of curves. In the context of conics, we first find the first derivative \(\frac{\mathrm{d}y}{\mathrm{d}x}\) by using implicit differentiation. Then, we differentiate again to find the second derivative \(\frac{\mathrm{d^2}y}{\mathrm{d}x^2}\). This process helps identify the properties of the conic, such as turning points and behavior at edges. It's essential to understanding how conics behave and is important for determining the order of the differential equation, which influences how we study and solve these equations.

General Equation of Conic

The general equation of a conic section without any rotation or translation is crucial for examining the conic's properties. In simple form, it shows as:

• \(Ax^2 + Bxy + Cy^2 = 0\)

This equation allows us to explore various conics by changing the values of \(A\), \(B\), and \(C\):

• \(A = C\) and \(B = 0\) indicates a circle.
• \(A eq C\) represents an ellipse (if \(A, C > 0\)) or a hyperbola (if \(A, C < 0\)).
• A single variable term can signify a parabola.

This versatile equation forms the starting point for deriving complex solutions and understanding complex geometrical shapes. Recognizing the types of conics from this equation is fundamental for students learning calculus and geometry.