The functions y=x2+cx2 are all solutions of equation: xy′+2y=4×2, (x>0). Find the constant c which produces a solution which also satisfi

Question

The functions y=x2+cx2 are all solutions of equation: xy′+2y=4×2, (x>0). Find the constant c which produces a solution which also satisfies the initial condition y(4)=4.

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Arya 2 weeks 2021-11-13T21:14:48+00:00 1 Answer 0 views 0

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    2021-11-13T21:15:48+00:00

    Answer:

    The required value of c is -192.

    Step-by-step explanation:

    We are given the differential equation:

    xy'+2y=4x^2 \\(x>0)

    The solution to the given differential equation is:

    y=x^2+\dfrac{c}{x^2}

    Initial condition:

    y(4) = 4

    Putting the values, in the solution, we get,

    4=(4)^2+\dfrac{c}{(4)^2}\\\\4 = 16+\dfrac{c}{(4)^2}\\\\\Rightarrow \dfrac{c}{(4)^2}=-12\\\Rightarrow c =-12(16)\\\Rightarrow c =-192

    Thus, the required value of c is -192.

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