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Math Help - Solve in two different ways (2)

  1. #1
    MHF Contributor alexmahone's Avatar
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    Solve in two different ways (2)

    Solve in two different ways.

    \frac{dy}{dx}=\frac{\sqrt{y}-y}{\tan x}

    Could someone please check the finer details of my solution, such as signs of integration constants and modulus signs?

    Separable:

    \frac{dy}{\sqrt{y}-y}=\frac{dx}{\tan x}

    \frac{1}{2\sqrt{y}}\frac{2}{1-\sqrt{y}}dy=\cot xdx

    \ln |\sin x|=\int \frac{1}{2\sqrt{y}}\frac{2}{1-\sqrt{y}}dy

    Let u=\sqrt{y}.

    du=\frac{dy}{2\sqrt{y}}

    \int \frac{1}{2\sqrt{y}}\frac{2}{1-\sqrt{y}}dy=\int \frac{2}{1-u}du

    =-2\ln |1-u|+C_1

    =-2\ln |1-\sqrt{y}|+C_1

    \ln |\sin x|=-2\ln |1-\sqrt{y}|+C_1

    \ln |\sin x|(1-\sqrt{y})^2=C_1

    |\sin x|(1-\sqrt{y})^2=e^{C_1}=C_2, where C_2 is a positive constant.

    (1-\sqrt{y})^2=C_2|\sin x|^{-1}

    1-\sqrt{y}=\pm \sqrt{C_2}|\sin x|^{-1/2}

    \sqrt{y}=1\mp \sqrt{C_2}|\sin x|^{-1/2}

    \sqrt{y}=1+C|\sin x|^{-1/2}, where C=\mp \sqrt{C_2}.

    y(x)=1+C^2|\sin x|^{-1}+2C|\sin x|^{-1/2}

    Bernoulli:

    \frac{dy}{dx}+\frac{y}{\tan x}=\frac{\sqrt{y}}{\tan x}

    Let u=y^{1/2}.

    y=u^2

    \frac{dy}{dx}=2u\frac{du}{dx}

    2u\frac{du}{dx}+\frac{u^2}{\tan x}=\frac{u}{\tan x}

    \frac{du}{dx}+\frac{u}{2\tan x}=\frac{1}{2\tan x}

    \frac{du}{dx}+\frac{u\cot x}{2}=\frac{\cot x}{2}

    The integrating factor is \exp\left(\int \frac{\cot x}{2}\right)=\exp\left(\frac{1}{2}\ln |\sin x|\right)=|\sin x|^{1/2}.

    |\sin x|^{1/2}\frac{du}{dx}+|\sin x|^{1/2}\frac{u\cot x}{2}=|\sin x|^{1/2}\frac{\cot x}{2}

    \frac{d}{dx}(u|\sin x|^{1/2})=|\sin x|^{1/2}\frac{\cot x}{2}

    u|\sin x|^{1/2}=|\sin x|^{1/2}+C

    u=1+C|\sin x|^{-1/2}

    y^{1/2}=1+C|\sin x|^{-1/2}

    y(x)=1+C^2|\sin x|^{-1}+2C|\sin x|^{-1/2}
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  2. #2
    A Plied Mathematician
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    Re: Solve in two different ways (2)

    The beauty of DE's is it's often much easier to check your own solution than to find the solution in the first place. Just plug your solution in to the DE and see if you get equality! You got the same answer both ways, which is a good sign.
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  3. #3
    MHF Contributor alexmahone's Avatar
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    Re: Solve in two different ways (2)

    Quote Originally Posted by Ackbeet View Post
    The beauty of DE's is it's often much easier to check your own solution than to find the solution in the first place. Just plug your solution in to the DE and see if you get equality! You got the same answer both ways, which is a good sign.
    How would I differentiate |\sin x|^{-1} and |\sin x|^{-1/2}.

    Also, are the modulus signs really necessary?
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    Re: Solve in two different ways (2)

    Note that

    \frac{d}{dx}\,|x|=\frac{x}{|x|}=\text{sgn}(x),

    the signum function.

    You can use the chain rule multiple times to get the final result.
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  5. #5
    MHF Contributor alexmahone's Avatar
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    Re: Solve in two different ways (2)

    Quote Originally Posted by Ackbeet View Post
    Note that

    \frac{d}{dx}\,|x|=\frac{x}{|x|}=\text{sgn}(x),

    the signum function.

    You can use the chain rule multiple times to get the final result.
    \frac{dy}{dx}=C^2*-1*|\sin x|^{-2}sgn(\sin x)\cos x +2C*-\frac{1}{2}|\sin x|^{-3/2}sgn(\sin x)\cos x

    =-C^2|\sin x|^{-2}sgn(\sin x)\cos x-C|\sin x|^{-3/2}sgn(\sin x)\cos x

    \frac{\sqrt{y}-y}{\tan x}=\frac{-C^2|\sin x|^{-1}-C|\sin x|^{-1/2}}{\tan x}

    =\frac{(-C^2|\sin x|^{-1}-C|\sin x|^{-1/2})\cos x}{\sin x}

    =\frac{(-C^2|\sin x|^{-2}-C|\sin x|^{-3/2})\cos x}{\sin x|\sin x|^{-1}} (Multiplying numerator and denominator by |\sin x|^{-1}.)

    =\frac{(-C^2|\sin x|^{-2}-C|\sin x|^{-3/2})\cos x}{sgn(\sin x)}

    =(-C^2|\sin x|^{-2}-C|\sin x|^{-3/2})\cos xsgn(\sin x)

    =-C^2|\sin x|^{-2}sgn(\sin x)\cos x-C|\sin x|^{-3/2}sgn(\sin x)\cos x

    \frac{dy}{dx}=\frac{\sqrt{y}-y}{\tan x}

    QED
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