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30 May, 12:09

Solve the equation. StartFraction dy Over dx EndFraction equals5 x Superscript 4 Baseline (1 plus y squared) Superscript three halves An implicit solution in the form F (x, y) equalsC is nothingequals C, where C is an arbitrary constant.

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  1. 30 May, 13:41
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    Step-by-step explanation:

    To solve the differential equation

    dy/dx = 5x^4 (1 + y²) ^ (3/2)

    First, separate the variables

    dy / (1 + y²) ^ (3/2) = 5x^4 dx

    Now, integrate both sides

    To integrate dy / (1 + y²) ^ (3/2), use the substitution y = tan (u)

    dy = (1/cos²u) du

    So,

    dy / (1 + y²) ^ (3/2) = [ (1/cos²u) / (1 + tan²u) ^ (3/2) ]du

    = (1/cos²u) / (1 + (sin²u/cos²u)) ^ (3/2)

    Because cos²u + sin²u = 1 (Trigonometric identity),

    The equation becomes

    [1 / (1/cos²u) ^ (3/2) * 1/cos²u] du

    = cos³u/cos²u

    = cosu

    Integral of cosu = sinu

    But y = tanu

    Therefore u = arctany

    We then have

    cos (arctany) = y/√ (1 + y²)

    Now, the integral of the equation

    dy / (1 + y²) ^ (3/2) = 5x^4 dx

    Is

    y/√ (1 + y²) = x^5 + C

    So

    y - (x^5 + C) √ (1 + y²) = 0

    is the required implicit solution
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