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Chapter 13: Problem 1104

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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Most popular questions from this chapter

Let \(\mathrm{m}\) and \(\mathrm{n}\) be respectively the degree and order of the differential equation of whose solution is \(\mathrm{y}=\mathrm{cx}+\mathrm{c}^{2}-3 \mathrm{c}^{(3 / 2)}+2\) where \(\mathrm{c}\) is parameter is (A) \(\mathrm{m}=1, \mathrm{n}=4\) (B) \(\mathrm{m}=1, \mathrm{n}=4\) (C) \(\mathrm{m}=2, \mathrm{n}=2\) (D) \(\mathrm{m}=4, \mathrm{n}=1\)

The solution of initial value problem \(\mathrm{x}(\mathrm{dy} / \mathrm{dx})=\mathrm{x}+\mathrm{y} ; \mathrm{y}(1)\) \(=1\) is \(\mathrm{y}=\) (A) \(x \log \overline{x-1}\) (B) \(x \log x+1\) (C) \(x(\log x+1)\) (D) none of these

If \((\mathrm{dy} / \mathrm{dx})=\mathrm{y}+3>0\) and \(\mathrm{y}(0)=2\), then \(\mathrm{y}(\log 2)\) is equal to. \(\begin{array}{ll}\text { (A) }-2 & \text { (B) } 5\end{array}\) (C) 7 (D) 13

The differential equation of the family of circles with fixed radius 5 units and centers on the line \(\mathrm{y}=2\) is: (A) \((\mathrm{y}-2)^{2}(\mathrm{dy} / \mathrm{dx})^{2}=25-(\mathrm{y}-2)^{2}\) (B) \((\mathrm{y}-2)(\mathrm{dy} / \mathrm{dx})^{2}=25-(\mathrm{y}-2)^{2}\) (C) \((x-2)(d y / d x)^{2}=25-(y-2)^{2}\) (D) \((\mathrm{x}-2)^{2}(\mathrm{dy} / \mathrm{dx})^{2}=25-(\mathrm{y}-2)^{2}\)

The solution of the equation \((2 x+y+1) d x+(4 x+2 y-1) d y\) \(=0\) is: (A) \(\log |2 \mathrm{x}+\mathrm{y}-1|+\mathrm{x}+2 \mathrm{y}=\mathrm{c}\) (B) \(\log (2 \mathrm{x}+\mathrm{y}+1)+\mathrm{x}+2 \mathrm{y}=\mathrm{c}\) (C) \(\log |2 \mathrm{x}+\mathrm{y}-1|=\mathrm{c}+\mathrm{x}+\mathrm{y}\) (D) \(\log (4 x+2 y-1)=c+2 x+y\)
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