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Math Help - Ln Equations Formation

  1. #1
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    Ln Equations Formation

    I need help with these four questions (If possible, can it be explained in steps):

    1.
    \sin x \frac{\mathrm{d}y}{\mathrm{d}x} = e^y, 0<x<\pi, y=0 at   x=\frac{\pi}{2}

    2.
    \sin x \frac{\mathrm{d}y}{\mathrm{d}x} = \tan y (3\cos x + \sin x),  y=\frac{\pi}{6} at x=\frac{\pi}{2}

    3.
    (5-3\sin x) \frac{\mathrm{d}y}{\mathrm{d}x} = 40\cos x,  y=0 at x=\frac{3\pi}{2}

    4.
    \frac{1}{y} \frac{\mathrm{d}y}{\mathrm{d}x} = x + xy,  y=1 at x=0
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  2. #2
    MHF Contributor red_dog's Avatar
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    1) \displaystyle\frac{dy}{e^y}=\frac{dx}{\sin x}
    \displaystyle\int\frac{dy}{e^y}=\int\frac{dx}{\sin x}

    2) \displaystyle\frac{dy}{\tan y}=\frac{3\cos x+\sin x}{\sin x}dx
    \displaystyle\int\cot ydy=\int(1+3\cot x)dx

    3) \displaystyle\frac{dy}{y(y+1)}=xdx
    \displaystyle\int\frac{dy}{y(y+1)}=\int xdx

    Now, integrate.
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