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Separation of Variables

AP Calculus AB· difficulty 3/5

Solve dy/dx=2xy2dy/dx = 2 x y^2 with y(0)=1y(0) = 1.

  • A

    y=1+x2y = 1 + x^2

  • B

    y=1−x2y = 1 - x^2

  • C

    y=11−x2y = \dfrac{1}{1 - x^2}

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  • D

    y=ex2y = e^{x^2}

Explanation

Separate: dy/y2=2x dxdy/y^2 = 2x\,dx. Integrate: −1/y=x2+C-1/y = x^2 + C. IC: −1=C-1 = C. So −1/y=x2−1⇒y=1/(1−x2)-1/y = x^2 - 1 \Rightarrow y = 1/(1 - x^2).

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