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Jared
Jared asked in Science & MathematicsMathematics · 9 years ago

First order nonlinear differential equation help?

First order nonlinear differential equation help?

i need help solving this equation please im stumped. included any step also. thanks

5y^4 y'= xy'+y

1 Answer

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  • Rapidfire
    Lv 7
    9 years ago
    Favorite Answer

    Rewrite this differential equation by rearranging then flipping the derivative:

    5y⁴y'= xy' + y

    5y⁴(dy / dx) = x(dy / dx) + y

    5y⁴(dy / dx) - x(dy / dx) = y

    (5y⁴ - x)(dy / dx) = y

    dy / dx = y / (5y⁴ - x)

    dx / dy = (5y⁴ - x) / y

    dx / dy = 5y³ - x / y

    Solve this differential equation by using an integrating factor:

    dx / dy + x / y = 5y³

    dx / dy + P(y)x = f(y)

    P(y) = 1 / y

    f(y) = 5y³

    I(y) = ℮^[∫ P(y) dy]

    I(y) = ℮^(∫ 1 / y dy)

    I(y) = ℮^(ln|y|)

    I(y) = y

    xy = ∫ I(y)f(y) dy

    xy = ∫ 5y⁴ dy

    xy = y⁵ + C

    x = y⁴ + C / y

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