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Math Help - Linear first order differential equations

  1. #1
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    Linear first order differential equations

    I need to solve the equation:

    x (dy/dx) + 5y = 7x where y(1) = 7/2

    The correct answer i have recieved is y = 7/6 (x + 2x^-5)

    How do i get to this answer?
    Ive been given this as a hint:

    x (dy/dx) + ay = bx^n can be simplified by multiplying both sides by x^(a-1)

    To give x^a (dy/dx) + a(x^(a-1))y = bx^(n+a-1)

    Which can be written as d/dx((x^a)y)

    But this doesnt seem to help me? Thanks
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  2. #2
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    I would divide both sides by \displaystyle x giving

    \displaystyle y' +\frac{5y}{x}=7

    Now find an integrating factor \displaystyle e^{\int \frac{5}{x}~dx}

    Multiply this guy through the equation and everything should start to piece together...
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  3. #3
    A Plied Mathematician
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    The equation is also Cauchy-Euler. You could assume a solution of the form y=x^{m}, and plug it into the DE to solve for m.

    Cheers.
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  4. #4
    Forum Admin topsquark's Avatar
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    It's also separable.
    \displaystyle x~\frac{dy}{dx} + 5y = 7x \implies \frac{dy}{7 - 5y} = \frac{dx}{x}

    -Dan
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  5. #5
    A Plied Mathematician
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    Reply to topsquark: I think you dropped the x multiplying the 7, which might change your result.
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