Solve a homogeneous equation by putting y = vx

1. A.
   $\tan^{-1}\!\left(\frac{y}{x}\right) + \log\sqrt{y^2+3x^2} \cdot x^{-2} = \log x + C$
2. B.
   $\frac{1}{\sqrt{3}}\tan^{-1}\!\left(\frac{y}{\sqrt{3}x}\right) + \log\left(\frac{\sqrt{y^2+3x^2}}{x}\right)^{1/2} = \log x + C$
3. C.
   $\frac{1}{\sqrt{3}}\tan^{-1}\!\left(\frac{y}{\sqrt{3}x}\right) - \log\left(\frac{\sqrt{y^2+3x^2}}{x}\right)^{1/2} = \log x + C$
4. D.
   $\tan^{-1}\!\left(\frac{x}{y}\right) + \log\sqrt{y^2+3x^2} \cdot x^{-2} = \log x + C$

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1. A.
   $\log\dfrac{x}{y} = cy$
2. B.
   $\log\dfrac{x}{y} = cx$
3. C.
   $\log\dfrac{y}{x} = cy$
4. D.
   $\log\dfrac{y}{x} = cx$

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1. A.
   $\frac{x^2+y^2}{x^3} = 1$
2. B.
   $x^2 + y^2 = x$
3. C.
   $x^2 + y^2 = x^4$
4. D.
   $x^2 + 2y^2 = x^4$

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1. A.
   $\log\frac{x}{y} = cy$, where $c$ is the constant of integration
2. B.
   $\log\frac{y}{x} = cy$, where $c$ is the constant of integration
3. C.
   $\log\frac{x}{y} = cx$, where $c$ is the constant of integration
4. D.
   $\log\frac{y}{x} = cx$, where $c$ is the constant of integration

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1. A.
   $2x^{2}\log x + y^{2} + 2cx^{2} = 0$, where c is the constant of integration
2. B.
   $2x^{2}\log x - y^{2} + 2cx^{2} = 0$, where c is the constant of integration
3. C.
   $x^{2}\log x + y^{2} + 2cx^{2} = 0$, where $c$ is the constant of integration
4. D.
   $x^{2}\log x - y^{2} + 2cxx^{2} = 0$, where c is the constant of integration

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1. A.
   $c(x^2 + y^2)^{\frac{1}{2}} + e^{\tan^{-1}\left(\frac{y^2}{x}\right)} = 0$, where c is an arbitrary constant
2. B.
   $c(x^2 + y^2)^{\frac{1}{2}} = e^{\tan^{-1}\left(\frac{y}{x}\right)}$, where c is an arbitrary constant
3. C.
   $c(x^2 - y^2) = e^{\tan^{-1}\left(\frac{y}{x}\right)}$, where c is an arbitrary constant
4. D.
   $c(x^2 + y^2) = e^{\tan^{-1}\left(\frac{y}{x}\right)}$, where c is an arbitrary constant

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1. A.
   $e$
2. B.
   $\pm\sqrt{3}\,e$
3. C.
   $3e^2$
4. D.
   $e^2$

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1. A.
   Circle
2. B.
   Ellipse
3. C.
   Hyperbola
4. D.
   Pair of straight Lines

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1. A.
   $\dfrac{y}{x} + \log|y| = c$
2. B.
   $\dfrac{y}{x} + \log|x| = c$
3. C.
   $\dfrac{y^2}{x} + \log|y| = c$
4. D.
   $\dfrac{y}{x^2} + \log|y| = c$

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1. A.
   $\log|y|$
2. B.
   $\tan\sqrt{y}$
3. C.
   $\cos 2y$
4. D.
   $3y^2 + 1$

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