Separation of Variables Practice

Sample questions

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1. Sample 1difficulty 2/5

   Which DE is separable?

   • A

     $dy/dx = x + y$

   • B

     $dy/dx = e^x \sin y$

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   • C

     $dy/dx = \cos(x + y)$

   • D

     $dy/dx = x y + 1$

   Why

   $dy/\sin y = e^x\,dx$ — separates into product of $x$-only and $y$-only terms.

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2. Sample 2difficulty 2/5

   For $dy/dx = xy$, separate variables:

   • A

     $dy/y = x\,dx$

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   • B

     $dy = xy\,dx$

   • C

     $dx/dy = y/x$

   • D

     $y\,dy = x\,dx$

   Why

   $dy/y = x\,dx$.

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3. Sample 3difficulty 3/5

   Solve $dy/dx = xy$ ($y > 0$).

   • A

     $y = x^2/2$

   • B

     $y = e^{x^2/2 + C}$

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   • C

     $y = e^{x^2}$

   • D

     $y = \ln(x^2/2)$

   Why

   $\ln y = x^2/2 + C \Rightarrow y = A e^{x^2/2}$ where $A = e^C$.

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4. Sample 4difficulty 3/5

   Solve $dy/dx = -y$ with $y(0) = 4$.

   • A

     $y = 4 e^x$

   • B

     $y = 4 - x^2$

   • C

     $y = 4 e^{-x}$

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   • D

     $y = -4 x + 4$

   Why

   $y = C e^{-x}$. $y(0) = C = 4$.

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5. Sample 5difficulty 3/5

   Solve $dy/dx = 2 x y^2$ with $y(0) = 1$.

   • A

     $y = 1 + x^2$

   • B

     $y = 1 - x^2$

   • C

     $y = \dfrac{1}{1 - x^2}$

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   • D

     $y = e^{x^2}$

   Why

   Separate: $dy/y^2 = 2x\,dx$. Integrate: $-1/y = x^2 + C$. IC: $-1 = C$. So $-1/y = x^2 - 1 \Rightarrow y = 1/(1 - x^2)$.

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