Solve a variable-separable equation (general or particular solution)

1. A.
   $y + \log y + e^x\cos^2 x = 2$
2. B.
   $\log(y+1) + e^x\cos^2 x = 1$
3. C.
   $y + \log y = e^x\cos^2 x$
4. D.
   $(y+1) + e^x\cos^2 x = 2$

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

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1. A.
   $y + x^2\tan^{-1}\frac{x}{2} + c = 0$, where c is a constant of integration.
2. B.
   $y + x\tan^{-1}x + c = 0$, where c is a constant of integration.
3. C.
   $y - x - \tan^{-1}x + c = 0$, where c is a constant of integration.
4. D.
   $y = \frac{x^2\tan^{-1}x}{2} - \frac{1}{2}x + \tan^{-1}x + c$, where c is constant of integration.

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1. A.
   $\log x - \frac{1}{2}(\log x)^2 - \tan^{-1}y = -\frac{\pi}{4}$
2. B.
   $\log x + \frac{1}{2}(\log x)^2 + \tan^{-1}y = \frac{\pi}{4}$
3. C.
   $\log x - \frac{1}{2}(\log x)^2 + \tan^{-1}y = \frac{\pi}{4}$
4. D.
   $\log x + \frac{1}{2}(\log x)^2 - \tan^{-1}y = \frac{\pi}{4}$

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1. A.
   $e^y \log y = e^x \sin x + c$
2. B.
   $e^y = e^x \sin x + c$
3. C.
   $\log y = e^y \sin x + c$
4. D.
   $y\log y = e^x \sin x + c$

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1. A.
   $y = e^{-3x}+c$
2. B.
   $y = e^x+c$
3. C.
   $y = e^{3x}+c$
4. D.
   $y = e^{-x}+c$

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1. A.
   $(1+\cos x)(1+\sin y)=c$
2. B.
   $1+\sin x+\cos y=c$
3. C.
   $(1+\sin x)(1+\cos y)=c$
4. D.
   $1+\sin x\cdot\cos y=c$

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1. A.
   x+y = c, where c is a constant of integration.
2. B.
   x-y = c(xy), where c is a constant of integration.
3. C.
   x+y = c(1+xy), where c is a constant of integration.
4. D.
   y-x = c(1+xy), where c is a constant of integration.

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1. A.
   $\frac{e^y}{y} = e^x \sin x + c$
2. B.
   $e^y \log y - e^x \cos x + c$
3. C.
   $e^y \log y = e^x \sin x + c$
4. D.
   $e^{y}\,y = e^{x}\sin x + c$

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1. A.
   $(1+\cos x)(1+\sin y)=c$, where $c$ is a constant of integration.
2. B.
   $1+\sin x+\cos y=c$, where $c$ is a constant of integration.
3. C.
   $(1+\sin x)(1+\cos y)=c$, where $c$ is a constant of integration.
4. D.
   $1+\sin x\cos y=c$, where $c$ is a constant of integration.

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1. A.
   $y = e^{- e^{2^{2}}}c$, where c is the constant of integration
2. B.
   $y = e^{- x^{2}}c$, where c is the constant of integration
3. C.
   $y = e^{e^{x^{2}}}c$, where $c$ is the constant of integration
4. D.
   $y = e^{x^{2}}c$, where $c$ is the constant of integration

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1. A.
   $\ \tan y = \left( \frac{1 - e^{3}}{1 - e^{x}} \right)^{3}$
2. B.
   $\ \tan y = \left( \frac{1 - e^{2}}{1 - e^{x}} \right)^{3}$
3. C.
   $\ \tan y = \left( \frac{1 - e}{1 - e^{x}} \right)^{3}$
4. D.
   $\ \tan y = \left( \frac{1 - e^{x}}{1 - e} \right)^{3}$

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1. A.
   $\cos x = ccosecy$, where c is the constant of integration.
2. B.
   $\sin x = c\sec y$, where $c$ is the constant of integration.
3. C.
   $\sin x = x\cos y$, where c is the constant of integration.
4. D.
   $\cos x = csiny$, where c is the constant of integration.

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

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1. A.
   $\log y + \dfrac{1}{y} + e^x\cos^2 x = 1$
2. B.
   $\log y + y + e^x\cos^2 x = 2$
3. C.
   $(y + 1) + e^x\cos^2 x = 2$
4. D.
   $\log\left(y + \dfrac{1}{y}\right) + e^x\cos^2 x = 1$

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1. A.
   $y - x = \log\left(\dfrac{x}{y}\right) + c$
2. B.
   $x - y = \log\left(\dfrac{x}{y}\right) + c$
3. C.
   $y + x = \log\left(\dfrac{x}{y}\right) + c$
4. D.
   $y + x = \log\left(\dfrac{y}{x}\right) + c$

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