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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 \((\mathrm{y} / \mathrm{x}) \cos (\mathrm{y} / \mathrm{x})[(\mathrm{dy} / \mathrm{dx})-(\mathrm{y} / \mathrm{x})]\) \(+\sin (\mathrm{y} / \mathrm{x})[(\mathrm{dy} / \mathrm{dx})+(\mathrm{y} / \mathrm{x})]=0 ; \mathrm{y}(1)=(\pi / 2)\) 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

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 differential equation \(x \sin (y / x) d y=(y \sin (y / x)-x) d x\) is: (A) \(\log \mathrm{y}=\cos (\mathrm{y} / \mathrm{x})+\mathrm{c}\) (B) \(\log \mathrm{x}=\cos (\mathrm{x} / \mathrm{y})+\mathrm{c}\) (C) \(\log \mathrm{x}=\cos (\mathrm{y} / \mathrm{x})+\mathrm{c}\) (D) \(\log \mathrm{y}=\cos (\mathrm{x} / \mathrm{y})+\mathrm{c}\)

Differential equation of the curves having the subnormal with \((7 / 2)\) units and passes through \((0,0)\) is: (A) \(x^{2}=7 y\) (B) \(\mathrm{y}^{2}=7 \mathrm{x}+\mathrm{c}, \mathrm{c} \neq 0\) (C) \(y^{2}=7 x\) (D) None of these

Let \(\mathrm{m}\) and \(\mathrm{n}\) be respectively the degree and order of the differential equation of whose solution is \(\mathrm{y}=\mathrm{cx}+\mathrm{c}^{2}-3 \mathrm{c}^{(3 / 2)}+2\) where \(\mathrm{c}\) is parameter is (A) \(\mathrm{m}=1, \mathrm{n}=4\) (B) \(\mathrm{m}=1, \mathrm{n}=4\) (C) \(\mathrm{m}=2, \mathrm{n}=2\) (D) \(\mathrm{m}=4, \mathrm{n}=1\)
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