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Chapter 13: Problem 1097

If \(\mathrm{m}\) and \(\mathrm{n}\) are order and degree of the equation \(\left(d^{2} y / d x^{2}\right)^{5}+4\left[\left(d^{2} y / d x^{2}\right)^{3} /\left(d^{3} y / d x^{3}\right)\right]+\left(d^{3} y / d x^{3}\right)=x^{2}-1\) then: (A) \(\mathrm{m}=3, \mathrm{n}=2\) (B) \(\mathrm{m}=3, \mathrm{n}=3\) (C) \(\mathrm{m}=3, \mathrm{n}=5\) (D) \(\mathrm{m}=3, \mathrm{n}=1\)

Short Answer

Expert verified
The order and degree of the given differential equation are 3 and 1, respectively. Thus, the correct answer is (D) \(\mathrm{m}=3, \mathrm{n}=1\).

Step by step solution

01

Identify the highest-order derivative

The equation contains the following derivative terms: - \(\left(\frac{d^2 y}{dx^2}\right)^5\) - \(\frac{\left(\frac{d^2 y}{dx^2}\right)^3}{\frac{d^3 y}{dx^3}}\) - \(\frac{d^3 y}{dx^3}\) The highest-order derivative in these terms is the third derivative \(\frac{d^3 y}{dx^3}\), as it has the highest number of derivatives.
02

Identify the highest power of the highest-order derivative

Now we'll examine the highest powers of the third derivative in the equation. In the second and third terms, the third derivative has power 1: - \(\frac{\left(\frac{d^2 y}{dx^2}\right)^3}{\frac{d^3 y}{dx^3}}\) has third derivative with power 1 in the denominator. - \(\frac{d^3 y}{dx^3}\) has third derivative with power 1. Since the highest power of the highest-order derivative is 1, the degree of the differential equation is 1.
03

Determine the order and degree

We have found that the order of the equation is 3 (due to the presence of third derivative) and the degree is 1 (as the highest power of the highest-order derivative is 1). Therefore, our answer is: (D) \(\mathrm{m}=3, \mathrm{n}=1\)

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Most popular questions from this chapter

\((\mathrm{dy} / \mathrm{dx})=\mathrm{e}^{\mathrm{x}+\mathrm{y}}+\mathrm{x}^{2} \mathrm{e}^{\mathrm{y}}\) has the particular solution for \(\mathrm{x}=\mathrm{y}\) \(=0:\) (A) \(\mathrm{e}^{\mathrm{x}}-\mathrm{e}^{-\mathrm{y}}+\left(\mathrm{x}^{3} / 3\right)=2\) (B) \(\mathrm{e}^{\mathrm{x}}+\mathrm{e}^{-\mathrm{y}}+\left(\mathrm{x}^{3} / 3\right)=2\) (C) \(\mathrm{e}^{\mathrm{x}-\mathrm{y}}+\left(\mathrm{x}^{3} / 3\right)=2\) (D) \(e^{y-x}-\left(x^{3} / 3\right)=2\)

Solution of \((\mathrm{y} / \mathrm{x}) \cos (\mathrm{y} / \mathrm{x})[(\mathrm{dy} / \mathrm{dx})-(\mathrm{y} / \mathrm{x})]\) \(+\sin (\mathrm{y} / \mathrm{x})[(\mathrm{dy} / \mathrm{dx})+(\mathrm{y} / \mathrm{x})]=0 ; \mathrm{y}(1)=(\pi / 2)\) is: (A) \(\mathrm{y} \sin (\mathrm{y} / \mathrm{x})=(\pi / 2 \mathrm{x})\) (B) \(\mathrm{y} \sin (\mathrm{y} / \mathrm{x})=(\pi / \mathrm{x})\) (C) \(\mathrm{y} \sin (\mathrm{y} / \mathrm{x})=(\pi / 3 \mathrm{x})\) (D) none of these

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

\(\underline{\text { Assertion - Reason Type Questions: }}\) Each question has four choices (a), (b), (c) and (d) out of which only one is correct. Write (a), (b), (c) and (d) according to the following rules. (a) Statement- 1 is True, Statement-2 is True, Statement- 2 is a correct explanation for Statement-1. (b) Statement-1 is True, Statement- 2 is True, Statement-2 is not a correct explanation for Statement-1. (c) Statement- 1 is True, Statement- 2 is False. (d) Statement- 1 is False, Statement- 2 is True. Statement \(-2:\) The differential equation \(\mathrm{y}^{\prime}=(\mathrm{y} / 2 \mathrm{x})\) is variable separable. Statement-1: Curve satisfying the differential equation \((\mathrm{dy} / \mathrm{dx})=(\mathrm{y} / 2 \mathrm{x})\) passing through \((2,1)\) is a parabola with Focus \([(1 / 4), \underline{0}]\). Statement- 2 : The differential equation \((\mathrm{dy} / \mathrm{dx})=(\mathrm{y} / 2 \mathrm{x})\) is variable separable.

Which of the following equations is a linear equation of order \(3 ?\) (A) \(\left(\mathrm{d}^{3} \mathrm{y} / \mathrm{dx}^{3}\right)+\left(\mathrm{d}^{2} \mathrm{y} / \mathrm{d} \mathrm{x}^{2}\right) \cdot(\mathrm{dy} / \mathrm{dx})+\mathrm{y}=\mathrm{x}\) (B) \(\left(\mathrm{d}^{3} \mathrm{y} / \mathrm{dx}^{3}\right)+\left(\mathrm{d}^{2} \mathrm{y} / \mathrm{dx}^{2}\right)+\mathrm{y}^{2}=\mathrm{x}^{2}\) (C) \(x \cdot\left(d^{3} y / d x^{3}\right)+\left(d^{3} y / d x^{3}\right)=e^{x}\) (D) \(\left(d^{2} y / d x^{2}\right)+(d y / d x)=\log x\)
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