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Chapter 13: Problem 1139

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

Short Answer

Expert verified
The solution of the given differential equation represents (C) a straight line passing through (0,0).

Step by step solution

01

Rewrite the Differential Equation

We can rewrite the given equation \(\mathrm{xdy}-\mathrm{ydx}=0\) as a separable equation: \(\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{y}{x}\)
02

Separate the Variables

Next, we separate the variables to get: \(\frac{\mathrm{dy}}{y} = \frac{\mathrm{dx}}{x}\)
03

Integrate Both Sides

Now, we integrate both sides: \(\int \frac{\mathrm{dy}}{y} = \int \frac{\mathrm{dx}}{x}\) Applying the integration, we get: \(\ln|y| = \ln|x | + C \), where \(C\) is the integration constant.
04

Simplify the Equation

Exponentiate both sides to remove the ln function: \(y = x \cdot e^C\) Since \(e^C\) is just a constant, let's call it \(k\). Therefore, our equation becomes: \(y = kx\)
05

Determine the Graphical Representation

The equation that we got, \(y = kx\), represents a straight line passing through the origin (0,0). So, the correct option is: (C) a straight line passing through (0,0)

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Most popular questions from this chapter

Solution of \(\left(d^{2} y / d x^{2}\right)=\log x\) is: (A) \(\mathrm{y}=(1 / 2) \mathrm{x}^{2} \log \mathrm{x}-(3 / 4) \mathrm{x}^{2}+\mathrm{c}_{1} \mathrm{x}+\mathrm{c}_{2}\) (B) \(\mathrm{y}=(1 / 2) \mathrm{x}^{2} \log \mathrm{x}+(3 / 4) \mathrm{x}^{2}+\mathrm{c}_{1} \mathrm{x}+\mathrm{c}_{2}\) (C) \(\mathrm{y}=(1 / 2) \mathrm{x}^{2} \log \mathrm{x}-(3 / 4) \mathrm{x}^{2}-\mathrm{c}_{1} \mathrm{x}+\mathrm{c}_{2}\) (D) None of these

The general solution of \([\mathrm{x}(\mathrm{dy} / \mathrm{dx})-\mathrm{y}] \mathrm{e}^{(\mathrm{y} / \mathrm{x})}=\mathrm{x}^{2} \cos \mathrm{x}\) is: (A) \(\mathrm{e}^{(\mathrm{x} / \mathrm{y})}=\cos \mathrm{x}+\mathrm{c}\) (B) \(\mathrm{e}^{(\mathrm{x} / \mathrm{y})}=\sin \mathrm{x}+\mathrm{c}\) (C) \(e^{(y / x)}=\sin x+c\) (D) \(e^{(y / x)}=\cos x+c\)

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) \((1 / 4) e^{-2 x}+c x+d\)

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 the differential equation \((\mathrm{dy} / \mathrm{dx})=(\mathrm{y} / \mathrm{x})+\left[\\{\Phi(\mathrm{y} / \mathrm{x})\\} /\left\\{\Phi^{1}(\mathrm{y} / \mathrm{x})\right\\}\right]\) is: (A) \(\phi(\mathrm{y} / \mathrm{x})=\mathrm{kx}\) (B) \(\Phi(\mathrm{y} / \mathrm{x})=\mathrm{ky}\) (C) \(\mathrm{x} \cdot \Phi(\mathrm{y} / \mathrm{x})=\mathrm{k}\) (D) \(\mathrm{y} \cdot \Phi(\mathrm{y} / \mathrm{x})=\mathrm{k}\)
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