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Chapter 13: Problem 1111

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) \(\log \cos x\)

Short Answer

Expert verified
\(P = \cot x\)

Step by step solution

01

Write down the given integrating factor and given equation

We are given that the integrating factor is \(\sin x\). The differential equation is given in the form: \[ \frac{dy}{dx} + Py = Q \]
02

Identify the form of the integrating factor

We know that the general formula for the integrating factor is \(e^{\int P dx}\). In this case, the integrating factor is given as \(\sin x\). So, we can rewrite the given integrating factor as: \[ \text{IF} = e^{\int P dx} = \sin x \]
03

Find P using the integrating factor formula

We want to find P such that the integrating factor is \(\sin x\). So we have the equation: \[ e^{\int P dx} = \sin x \] To solve for P, first, take the natural logarithm on both sides: \[ \int P dx = \ln(\sin x) + C \] Now, differentiate both sides with respect to x: \[ P = \frac{d}{dx}(\ln(\sin x)) \] Use the chain rule on the right side: \[ P = \frac{1}{\sin x} \cdot (\frac{d}{dx}(\sin x)) = \cot x \] So, based on our analysis, the correct value for P is \(\cot x\), which corresponds to answer choice (C).

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Most popular questions from this chapter

The degree and order of the differential equation of the family of all parabolas whose axis is \(\mathrm{x}\) -axis, are respectively. (A) 1,2 (B) 3,2 (C) 2,3 (D) 2,1

The solution of \(\mathrm{xdy}-\mathrm{ydx}=0\) represents: (A) parabola having vertex at \((0,0)\) (B) circle having centre at \((0,0)\) (C) a straight line passing through \((0,0)\) (D) a rectangular hyperbola

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) \(\log \mathrm{y}=\mathrm{cx}\)

The differential equation of all parabolas having the directrix parallel to \(\mathrm{x}\) -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) \(\left(\mathrm{d}^{3} \mathrm{y} / \mathrm{dx}^{3}\right)+\left(\mathrm{d}^{2} \mathrm{y} / \mathrm{dx}^{2}\right)=0\) (D) \(\left(d^{2} y / d x^{2}\right)=0\)

If \(\mathrm{x}(\mathrm{dy} / \mathrm{dx})=\mathrm{y}(\log \mathrm{y}-\log \mathrm{x}+1)\), then the solution of the equation is: (A) \(x \log (\mathrm{y} / \mathrm{x})=\mathrm{cy}\) (B) \(\log (\mathrm{y} / \mathrm{x})=\mathrm{cx}\) (C) \(\log (\mathrm{x} / \mathrm{y})=\mathrm{cy}\) (D) \(\mathrm{y} \cdot \log (\mathrm{x} / \mathrm{y})=\mathrm{cx}\)
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