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Math Help - Nonlinear 1st order ODE

  1. #1
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    Nonlinear 1st order ODE

    \displaystyle \frac{dy}{dx}=\cos(x+y)

    How can this be solved?
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  2. #2
    Master Of Puppets
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    Would it help if \cos(x+y)= \cos x \cos y - \sin x \sin y ??
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  3. #3
    Math Engineering Student
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    put t=x+y and the equation becomes separable.
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  4. #4
    Super Member Random Variable's Avatar
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    let  u = x+y

    then  y= u-x and  \frac{dy}{dx} = \frac{du}{dx} -1

    so  \frac{du}{dx} -1 = \cos u

     \frac{du}{1+\cos u} = dx

    integrate both sides

     \tan\Big(\frac{u}{2}\Big) = x + C (the substitution  v = \tan\Big(\frac{u}{2}\Big) transforms the integral on the left into  \int dv )


    then  u = 2 \arctan (x+C)

    and finally  y = 2 \arctan(x+C)-x
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  5. #5
    Math Engineering Student
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    \frac{1}{1+\cos u}=\frac{1-\cos u}{\sin ^{2}u}=\csc ^{2}(u)-\cot (u)\csc (u), faster to integrate.
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