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Chapter 13: Problem 1141

The solution of \(\left(\mathrm{d}^{2} \mathrm{y} / \mathrm{dx}^{2}\right)=0\) represents: (A) a point (B) a straight line (C) a parabola (D) a circle

Short Answer

Expert verified
The solution of the given equation \(\frac{d^2y}{dx^2} = 0\) represents a straight line, as integrating twice gives the equation \(y = C_1x + C_2\), which is the equation for a straight line. So, the correct answer is (B) a straight line.

Step by step solution

01

Integrate the given equation once

Given the second derivative of y with respect to x is 0: \(\frac{d^2y}{dx^2} = 0\) Integrate with respect to x: \(\frac{dy}{dx} = \int 0 \, dx = C_1\) Where \(C_1\) is the constant of integration.
02

Integrate the equation again

Now integrate the equation \(\frac{dy}{dx} = C_1\) with respect to x: \(y = \int C_1 \, dx = C_1x + C_2\) Where \(C_1\) and \(C_2\) are constants of integration.
03

Identify the shape of the solution

The equation we have obtained is in the form: \(y = C_1x + C_2\) This equation represents a straight line, hence the solution represents a straight line. So, the correct answer is (B) a straight line.

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Most popular questions from this chapter

Integrating factor of differential equation \((1+x)(d y / d x)-x \cdot y=1-x\) is: (A) \(1+\mathrm{x}\) (B) \(\log (1+x)\) (C) \(e^{-x}(1+x)\) (D) \(x \cdot e^{x}\)

The differential equation representing the family of curves \(\mathrm{y}^{2}=2 \mathrm{c}(\mathrm{x}+\mathrm{v} \mathrm{c})\), where \(\mathrm{c}\) is a positive parameter, is of order and degree as follows. (A) order 2, degree 1 (B) order 1 , degree 2 (C) order 2 , degree 2 (D) order 1 , degree 3

The differential equation of all parabolas having axis parallel to y-axis: (A) \(\left(\mathrm{d}^{3} \mathrm{x} / \mathrm{dy}^{3}\right)^{2}=0\) (B) \(\left(\mathrm{d}^{3} \mathrm{y} / \mathrm{dx}^{3}\right)=0\) (C) \(\left(\mathrm{d}^{3} \mathrm{y} / \mathrm{dx}^{3}\right)+\left(\mathrm{d}^{2} \mathrm{y} / \mathrm{d} \mathrm{x}^{2}\right)=0\) (D) \(\left(\mathrm{d}^{2} \mathrm{y} / \mathrm{dx}^{2}\right)=0\)

\(\underline{\text { Assertion - Reason Type Questions: }}\) 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.

The solution of the equation \(\left(d^{2} y / d x^{2}\right)=e^{-2 x}\) is: \(y=\) (A) \((1 / 4) \mathrm{e}^{-2 \mathrm{x}}+\mathrm{cx}+\mathrm{d}\) (B) \((1 / 4) e^{-2 \mathrm{x}}\) (C) \((1 / 4) e^{-2 x}+c x^{2}+d\) (D) \((1 / 4) e^{-2 x}+c x+d\)
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