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Chapter 13: Problem 1168

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) \(e^{(y / x)}=\cos x+c\)

Short Answer

Expert verified
The general solution is \(e^{(y / x)}=\sin x+c\).

Step by step solution

01

Identify the type of differential equation

This equation does not fit any standard form of the first order differential equations. Rearranging the given equation for a more intuitive form, we get: \[\dfrac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{x}} = \dfrac{\mathrm{y}\cdot \mathrm{e}^{(\mathrm{y}/\mathrm{x})}}{\mathrm{x}} + \mathrm{x} \cos \mathrm{x}\] However, on close examination, we see that the equation is separable. Therefore, we proceed to separate the variables.
02

Separating variables

To separate the variables, we need to bring all the terms involving `y` on the left side and those involving `x` on the right side: \[\dfrac{\mathrm{e}^{(\mathrm{y}/\mathrm{x})} \mathrm{d}\mathrm{y}}{\mathrm{y}} = \dfrac{\mathrm{x} \cos \mathrm{x} \mathrm{d}\mathrm{x}}{\mathrm{x}}\] The equation becomes: \[\mathrm{e}^{(\mathrm{y}/\mathrm{x})} \dfrac{\mathrm{d}\mathrm{y}}{\mathrm{y}} = \cos \mathrm{x} \cdot \mathrm{d} \mathrm{x}\]
03

Integrating both sides

Now we integrate both sides of the equation: \[\int \mathrm{e}^{(\mathrm{y}/\mathrm{x})} \dfrac{\mathrm{d}\mathrm{y}}{\mathrm{y}} = \int \cos \mathrm{x} \cdot \mathrm{d} \mathrm{x}\] On integrating, we get: \[\mathrm{e}^{(y/x)} = \sin x + C\] where 'C' is the integration constant. Now, let us compare this result with the given options.
04

Comparing with Options

Comparing the obtained result with the given options: (A) \(\mathrm{e}^{(\mathrm{x} / \mathrm{y})}=\cos \mathrm{x}+\mathrm{c}\) - Incorrect (B) \(\mathrm{e}^{(\mathrm{x} / \mathrm{y})}=\sin \mathrm{x}+\mathrm{c}\) - Incorrect (C) \(e^{(y / x)}=\sin x+c\) - Correct (D) \(e^{(y / x)}=\cos x+c\) - Incorrect Therefore, the correct general solution is: \[\boxed{e^{(y / x)}=\sin x+c}\]

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Most popular questions from this chapter

\(\mathrm{y}^{2}=(\mathrm{x}-\mathrm{c})^{3}\) is general solution of the differential equation: (where c is arbitrary constant). (A) \((\mathrm{dy} / \mathrm{dx})^{3}=27 \mathrm{y}\) (B) \(2(\mathrm{dy} / \mathrm{dx})^{3}-8 \mathrm{y}=0\) (C) \(8(\mathrm{dy} / \mathrm{dx})^{3}=27 \mathrm{y}\) (D) \(8\left(\mathrm{~d}^{3} \mathrm{y} / \mathrm{dx}^{3}\right)-27 \mathrm{y}=0\)

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 differential equation of the family of circles with fixed radius 5 units and centers on the line \(\mathrm{y}=2\) is: (A) \((\mathrm{y}-2)^{2}(\mathrm{dy} / \mathrm{dx})^{2}=25-(\mathrm{y}-2)^{2}\) (B) \((\mathrm{y}-2)(\mathrm{dy} / \mathrm{dx})^{2}=25-(\mathrm{y}-2)^{2}\) (C) \((x-2)(d y / d x)^{2}=25-(y-2)^{2}\) (D) \((\mathrm{x}-2)^{2}(\mathrm{dy} / \mathrm{dx})^{2}=25-(\mathrm{y}-2)^{2}\)

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

The order of the differential equation of family of circle touching a fixed straight line passing through origin is. (A) 2 (B) 3 (C) 4 (D) none of these
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