What is the general solution of the differential equation ? x dy/dx=2/x+2−y given that x=−1,y=0

Lina Neal 2022-09-11 Answered
What is the general solution of the differential equation ? x d y d x = 2 x + 2 - y given that x=−1,y=0
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Answers (1)

Adolfo Lee
Answered 2022-09-12 Author has 17 answers
We have:

x d y d x = 2 x + 2 - y ..... [A]

We can rearrange [A] as follows:

d y d x = 2 x 2 + 2 x - y x

d y d x + y x = 2 x 2 + 2 x ..... [B]

We can use an integrating factor when we have a First Order Linear non-homogeneous Ordinary Differential Equation of the form;

d y d x + P ( x ) y = Q ( x )

So we form an Integrating Factor;

I = e P ( x ) d x
    = exp (   1 x   d x )
    = exp ( ln x )
    = x

And if we multiply the DE [B] by this Integrating Factor, I, we will have a perfect product differential;

    x   d y d x + y = 2 x + 2 x

d d x ( x y ) = 2 x + 2

Which we can directly integrate to get:

x y =   2 x + 2 x   d x
        = 2 ln | x | + 2 x + C

Using the initial Condition, x=−1,y=0 we have:

0 = 2 ln | 1 | - 2 + C C = 2

Leading to the General Solution:

x y = 2 ln | x | + 2 x + 2
y = ln | x | + x + 1 x

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