Anonymous
Anonymous asked in Science & MathematicsMathematics · 1 month ago

Solve the given differential equation by separation of variables.?

Solve the given differential equation by separation of variables.

e^xy (dy/dx) = e^-y + e^(-5x-y)

Update:

UPDATE: 

separated the variables: ye^y dy = ((1/e^x)+e^-6x) dx

2 Answers

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  • Ian H
    Lv 7
    1 month ago

    e^(x)*e^(y)*(dy/dx) = e^(-y) + e^(-5x)*e^-y

    e^(x)*e^(2y)*(dy/dx) = 1 + e^(-5x)

    e^(2y)*(dy/dx) = e^(-x) + e^(-6x)

    e^(2y)dy = [e^(-x) + e^(-6x)] dx

    (1/2)e^(2y) = -[k + e^(-x) + (1/6)e^(-6x)]

    e^(2y) = -[2k + 2e^(-x) + (1/3)e^(-6x)]

    2y = ln{-[2k + 2e^(-x) + (1/3)e^(-6x)]}

    y = (1/2)ln[2e^(-x) + (1/3)e^(-6x) + C]

  • 1 month ago

    e^xy(dy/dx) = e^-y + (e^-5x)e^-y

    so, (e^x)(e^y)(dy/dx) = e^-y[1 + e^-5x]

    => (e^2y)(dy/dx) = (1 + e^-5x)/e^x

    => (e^2y)(dy/dx) = e^-x + e^-6x

    so, ∫ (e^2y) dy = ∫ (e^-x + e^-6x) dx

    i.e. (1/2)e^2y = -e^-x - (1/6)e^-6x + C

    so, 3e^2y = -6e^-x - e^-6x + D

    :)>

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