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Math Help - Solving differential equations...

  1. #1
    Newbie italipinogirl's Avatar
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    Solving differential equations...

    Hello, everyone. I am new here. This is just high school Calculus. Don't laugh at me .

    Can someone tell me how these problems work? I even looked at the answers (yes, I *am* allowed to do that) but I can't work them backwards. The first one is this:

    dy/dx = y+2

    And this is what I did:

    dy = (y+2) dx
    y = Ce^x + 2x

    Now the answer is y = Ce^x - 2 .

    And the second one: y' = y√x

    And last: (1 + x^2)y' = 2xy
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  2. #2
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    Re: Solving differential equations...

    Hello, italipinogirl!

    \frac{dy}{dx} \:=\: y+2

    And this is what I did: . dy \:=\: (y+2) dx \quad\Rightarrow\quad y\:=\: Ce^x + 2x
    . . How did you get 2x?

    Now the answer is: . y \:=\: Ce^x - 2

    We have: . \frac{dy}{dx} \:=\:y+2

    Separate variables: . \frac{dy}{y+2} \:=\:dx

    Integrate: . \int\frac{dy}{y+2} \:=\:\int dx \quad\Rightarrow\quad \ln|y+2| \:=\:x+c

    . . . . . . . . y+2 \:=\:e^{x+c} \:=\:e^x\cdot e^c \:=\:e^x\cdot C

    . . . . . . . . y+2 \:=\:Ce^x \quad\Rightarrow\quad y \:=\:Ce^x - 2




    \frac{dy}{dx} \:=\: y\sqrt{x}

    We have: . \frac{dy}{dx} \:=\:yx^{\frac{1}{2}}

    Separate variables: . \frac{dy}{y} \:=\:x^{\frac{1}{2}}dx

    Integrate: . \ln |y| \;=\;\tfrac{2}{3}x^{\frac{3}{2}}+c

    . . . . . . . . y \;=\;e^{\frac{2}{3}x^{\frac{3}{2}} + c} \;=\;e^{\frac{2}{3}x^{\frac{3}{2}}}\cdot e^c \;=\;e^{\frac{2}{3}x^{\frac{3}{2}}}\cdot C

    . . . . . . . . y \;=\;Ce^{\frac{2}{3}x^{\frac{3}{2}}}




    (1 + x^2)\frac{dy}{dx}\:=\: 2xy

    Separate variables: . \frac{dy}{y} \;=\;\frac{2x}{1+x^2}\,dx

    Integrate: . \ln(y) \;=\;\ln(1+x^2) + c \;=\;\ln(1+x^2) + \ln C \;=\;\ln[C(1+x^2)]

    Therefore: . y \;=\;C(1+x^2)

    Thanks from italipinogirl
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  3. #3
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    Re: Solving differential equations...

    First :
    dy/dx = y+2
    dy/(y+2) = dx
    integrate it.

    Second :
    dy/dx = y√x
    dy/y = (√x)dx
    integrate it.

    Third :
    (1 + x^2)(dy/dx) = 2xy
    dy/y = (2x/(1+x^2))dx
    integrate it.
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  4. #4
    Newbie italipinogirl's Avatar
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    Re: Solving differential equations...

    Wow, thank you! That completely makes sense now.
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