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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

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) \(\left(\mathrm{d}^{2} \mathrm{y} / \mathrm{dx}^{2}\right)=0\)

Family \(\mathrm{y}=\mathrm{Ax}+\mathrm{A}^{3}\) of curves is represented by the differential equation of degree: (A) 1 (B) 2 (C) 3 (D) 4

If \(\sin x\) is an Integrating factor of \((d y / d x)+p \cdot y=Q\) then \(p\) is: (A) \(\sin x\) (B) \(\log \sin x\) (C) \(\cot x\) (D) \(\log \cos x\)

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) \(\left(d^{2} y / d x^{2}\right)+(d y / d x)=\log x\)

The solution of the differential equation \(\mathrm{ydx}+\left(\mathrm{x}+\mathrm{x}^{2} \mathrm{y}\right) \mathrm{dy}=0\) is: (A) \((1 / \mathrm{xy})+\log \mathrm{y}=\mathrm{c}\) (B) \(-(1 / x y)+\log y=c\) (C) \(-(1 / \mathrm{xy})=\mathrm{c}\) (D) \(\log \mathrm{y}=\mathrm{cx}\)
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