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    Solutions to Several First Order ODEs

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    Solve 1st order DE, please include explaination of process, if possible.

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    https://brainmass.com/math/integrals/solutions-several-first-order-odes-428597

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    1. We have

    dy/dx = (x + 3y)/(3x + y) = (1 + 3y/x)/(3 + y/x).

    Let u = y/x, so y = ux and dy/dx = u + x du/dx. Then we have

    u + x du/dx = (1 + 3u)/(3 + u)

    so

    x du/dx = (1 + 3u)/(3 + u) - u
    = (1 - u^2)/(3 + u).

    Thus we have

    (3 + u)/(1 - u^2) du = dx/x.

    Integrating both sides we find

    ln((1 + u)/(1 - u)^2) = ln x + C1.

    Exponentiating both sides we have

    (1 + u)/(1 - u)^2 = Cx.

    Substituting y back into the equation and simplifying, we find

    x + y = C(x - y)^2.

    2. We have

    -y dx + (x + sqrt(xy)) dy = 0,

    whence

    dy/dx = y/(x + sqrt(xy)) = u/(1 + sqrt(u))

    where u = y/x. Thus we have

    u + x du/dx = u/(1 + sqrt(u))

    whence

    x du/dx = u/(1 + sqrt(u)) - u
    = u(1 - 1 - sqrt(t))/(1 + sqrt(u))
    = -u sqrt(u)/(1 + sqrt(u)).

    Thus we have

    (1 + u^(1/2)) / u^(3/2) du = dx/x
    [u^(-3/2) + u^-1] du = dx/x.

    Integrating both sides we find

    -2 u^(-1/2) + ln u = ln x + C.

    Substituting y back into this equation, we have

    -2 sqrt(x/y) + ln y - ln x = ln x + C

    whence

    -2 sqrt(x/y) + ln y - 2 ln x = C.

    3. We have

    dy/dx = y/x + x/y

    which is clearly homogeneous. Substituting u = y/x, we find

    u + x du/dx = u + 1/u

    whence

    x du/dx = 1/u

    from which it follows that

    u du = dx/x.

    Integrating both sides, we find

    1/2 u^2 = ln x + C.

    Substituting y back in, we find

    y^2/(2x^2) = ln x + C

    whence

    y^2 = 2x^2 ln x + C

    so the solution to the DE is

    y = +/- sqrt(2x^2 ln x + C).

    4. We have

    (y + x cot(y/x)) dx - x dy = 0,

    from which it follows that

    (y/x + (x/y) cot(x/y)) dx - dy = 0,

    which is clearly homogeneous. From the substitution u = y/x, we have

    (u + (cot u)/u) dx - (u dx + x du) = 0,

    from which we obtain

    (cot u)/u dx = x du,

    from which ...

    Solution Summary

    We use various methods to solve several first order ordinary differential equations.

    $2.49

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