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

Short Answer

Expert verified
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

01

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.
02

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: \[\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{-2Ax - By}{B + 2Cy}\]
03

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 \(\frac{\mathrm{d^2}y}{\mathrm{d}x^2}\): \[(B+2Cy)\frac{\mathrm{d^2}y}{\mathrm{d}x^2} + (2A)(2) = 0\]
04

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.
05

Match the result with the given options

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

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