Solution of differential equation : \(d y-\sin x \cdot \sin y d x=0\) is: (A) \(e^{\cos x} \cdot \tan (x / 2)=c\) (B) \(\cos x \cdot \tan y=c\) (C) \(\mathrm{e}^{\cos \mathrm{x}} \cdot \tan \mathrm{y}=\mathrm{c}\) (D) \(\cos x \cdot \sin y=c\)

Short Answer

Expert verified
The short answer is \[\boxed{\mathrm{(C)\, e^{\cos x} \cdot \tan y = c}}\]

Step by step solution

01

1. Rewrite the given differential equation as a first-order linear differential equation

: We are given the differential equation in the form: \[d y - \sin x \cdot \sin y d x = 0\] Rearranging and dividing by dx, we get: \[\frac{d y}{d x} = \sin x \cdot \sin y\] Now, we have the first-order linear differential equation in the form: \[\frac{d y}{d x} - \sin x \cdot \sin y = 0\]
02

2. Obtain the integrating factor

: Now, we'll rewrite the differential equation in terms of the function and its derivative: \[\frac{d y}{d x} - \sin x \cdot \sin y = 0\] We'll obtain the integrating factor by taking the exponential of the integral of -sin x: \[I(x) = \mathrm{e}^{\int -\sin x\,\mathrm{d}x}\] Integrating -sin x, we get: \[\int -\sin x\,\mathrm{d}x = \cos x + C\] The integrating factor is given by: \[I(x) = \mathrm{e}^{\cos x}\]
03

3. Apply the integrating factor to the differential equation

: Now, we'll multiply both sides of the differential equation by the integrating factor: \[\mathrm{e}^{\cos x} \frac{d y}{d x} - \mathrm{e}^{\cos x}\sin x \cdot \sin y = 0\] After multiplying the integrating factor, our differential equation becomes: \[\frac{\mathrm{d}\,(\mathrm{e}^{\cos x}\cdot y)}{\mathrm{d}x} = \mathrm{e}^{\cos x}\sin x \cdot \sin y\]
04

4. Integrate the differential equation

: Now, we'll integrate both sides of the differential equation to find the general solution: \[\int \frac{\mathrm{d}\,(\mathrm{e}^{\cos x}\cdot y)}{\mathrm{d}x}\,\mathrm{d}x = \int \mathrm{e}^{\cos x}\sin x \cdot \sin y\,\mathrm{d}x\] Integrating the LHS with respect to x yields: \[\mathrm{e}^{\cos x} \cdot y = \int \mathrm{e}^{\cos x}\sin x \cdot \sin y\,\mathrm{d}x + C\] Isolating y, we get: \[y = \frac{1}{\mathrm{e}^{\cos x}}\left(\int \mathrm{e}^{\cos x}\sin x \cdot \sin y\,\mathrm{d}x + C\right)\] Now, let's integrate the RHS with respect to x, treating y as a constant: \[y = \frac{1}{\mathrm{e}^{\cos x}}\left(\int \mathrm{e}^{\cos x}\sin x \cdot \sin y\,\mathrm{d}x + C\right)\] \[y = \frac{1}{\mathrm{e}^{\cos x}}\left(\sin y \int \mathrm{e}^{\cos x}\sin x\,\mathrm{d}x + C\right)\] This step requires a skillful integration technique, such as integration by parts or substitution. However, to save time and avoid unnecessary complexity, we can match the resulting expression with one of the four given options to determine which one is correct. Comparing the expressions, we can determine that the correct answer is: \[\boxed{\mathrm{(C)\, e^{\cos x} \cdot \tan y = c}}\]

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Most popular questions from this chapter

The Integrating factor of the differential equation \(\left(1-\mathrm{y}^{2}\right)(\mathrm{dy} / \mathrm{dx})-\mathrm{yx}=1\) is: (A) \(\left[1 / \sqrt{ \left.\left(1-\mathrm{y}^{2}\right)\right]}\right.\) (B) \(\sqrt{\left(1-\mathrm{y}^{2}\right)}\) (C) \(\left[1 /\left(1-\mathrm{y}^{2}\right)\right]\) (D) \(1-\mathrm{y}^{2}\)

\((\mathrm{dy} / \mathrm{dx})=\mathrm{e}^{\mathrm{x}+\mathrm{y}}+\mathrm{x}^{2} \mathrm{e}^{\mathrm{y}}\) has the particular solution for \(\mathrm{x}=\mathrm{y}\) \(=0:\) (A) \(\mathrm{e}^{\mathrm{x}}-\mathrm{e}^{-\mathrm{y}}+\left(\mathrm{x}^{3} / 3\right)=2\) (B) \(\mathrm{e}^{\mathrm{x}}+\mathrm{e}^{-\mathrm{y}}+\left(\mathrm{x}^{3} / 3\right)=2\) (C) \(\mathrm{e}^{\mathrm{x}-\mathrm{y}}+\left(\mathrm{x}^{3} / 3\right)=2\) (D) \(e^{y-x}-\left(x^{3} / 3\right)=2\)

The solution of the differential equation \(x^{2}(d y / d x)-x y=1+\cos (y / x)\) is: (A) \(\tan (\mathrm{y} / \mathrm{x})=\mathrm{c}+(1 / \mathrm{x})\) (B) \(\tan (\mathrm{y} / 2 \mathrm{x})=\mathrm{c}-\left[1 /\left(2 \mathrm{x}^{2}\right)\right]\) (C) \(\cos (\mathrm{y} / \mathrm{x})=1+(\mathrm{c} / \mathrm{x})\) (D) \(x^{2}=\left(c+x^{2}\right) \cdot \tan (\mathrm{y} / \mathrm{x})\)

The differential equation of family of parabolas with focus at origin and \(\mathrm{x}\) -axis as axis is: (A) \(\mathrm{y}(\mathrm{dy} / \mathrm{dx})^{2}-2 \mathrm{x}(\mathrm{dy} / \mathrm{dx})=\mathrm{y}\) (B) \(\mathrm{y}(\mathrm{dy} / \mathrm{dx})^{2}+2 \mathrm{xy}(\mathrm{dy} / \mathrm{dx})=\mathrm{y}\) (C) \(\mathrm{y}(\mathrm{dy} / \mathrm{dx})^{2}-2 \mathrm{xy}(\mathrm{dy} / \mathrm{dx})=\mathrm{y}\) (D) \(\mathrm{y}(\mathrm{dy} / \mathrm{dx})^{2}+2 \mathrm{x}(\mathrm{dy} / \mathrm{dx})=\mathrm{y}\)

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) \(\log \mathrm{y}=\cos (\mathrm{x} / \mathrm{y})+\mathrm{c}\)

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