Use an integrating factor to solve the following first order linear ODE. xy’ + 2y = 3x, y(1) = 3 Find the end behavior of y as x rightarrow

Question

Use an integrating factor to solve the following first order linear ODE. xy’ + 2y = 3x, y(1) = 3 Find the end behavior of y as x rightarrow infinity.

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Kylie 2 weeks 2021-09-12T08:27:39+00:00 1 Answer 0

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    2021-09-12T08:28:52+00:00

    Answer:

    Solution –>  y(x) = x + \frac{2}{x^{2}}

    when x –> infinity the y goes to infinity too.

    Step-by-step explanation:

    We have the eq.

     x y'+2 y = 3 x

    with y(1) = 3. So a reconfiguration o this last one equation can be like:

     y' + \frac{2}{x} y = 3

    so is of the form y’+p(x) y = f(x), where the integral factor is given by:

     \mu = \exp[ \int p(x) dx]

    is,

     \mu = \exp[ \int \frac{2}{x} dx] = x^{2}

    Multiplying the whole equation with this integral factor we can write the expression:

     \int \frac{d}{dx}[y x^{2}] dx = \int 3x^{2} dx

    and from this we obtain,

     y x^{2} = x^{3} + c,

    So when y(x=1) = 3, c = 2 and the complete solution can be writen as:

     y(x) = x - \frac{2}{x^{2}} .

    And we can see that when x –> infinity the second term of the solution is practically zero and the first is infinity so y also goes to infinity when x does.

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