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28 August, 00:44

If the right side of the equation dy dx = f (x, y) can be expressed as a function of the ratio y/x only, then the equation is said to be homogeneous. Such equations can always be transformed into separable equations by a change of the dependent variable. The following method outline can be used for any homogeneous equation. That is, the substitution y = xv (x) transforms a homogeneous equation into a separable equation. The latter equation can be solved by direct integration, and then replacing v by y x gives the solution to the original equation. dy/dx = (x^2 + 5y^2) / 2xy

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  1. 28 August, 01:22
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    y = x*sqrt (Cx - 1)

    Step-by-step explanation:

    Given:

    dy / dx = (x^2 + 5y^2) / 2xy

    Find:

    Solve the given ODE by using appropriate substitution.

    Solution:

    - Rewrite the given ODE:

    dy/dx = 0.5 (x/y) + 2.5 (y/x)

    - use substitution y = x*v (x)

    dy/dx = v + x*dv/dx

    - Combine the two equations:

    v + x*dv/dx = 0.5 * (1/v) + 2.5*v

    x*dv/dx = 0.5 * (1/v) + 1.5*v

    x*dv/dx = (v^2 + 1) / 2v

    -Separate variables:

    (2v. dv / (v^2 + 1) = dx / x

    - Integrate both sides:

    Ln (v^2 + 1) = Ln (x) + C

    v^2 + 1 = Cx

    v = sqrt (Cx - 1)

    - Back substitution:

    (y/x) = sqrt (Cx - 1)

    y = x*sqrt (Cx - 1)
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