What is a particular solution to the differential equation dy/dx=4√ylnx/x with y(e)=1?

Hugh Soto 2022-09-12 Answered
What is a particular solution to the differential equation d y d x = 4 y ln x x with y(e)=1?
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Answers (1)

Clarence Mills
Answered 2022-09-13 Author has 18 answers
We have:

d y d x = 4 y ln x x

Which is a first order linear separable Differential Equation, so we can rearrange to get:

1 y d y d x = 4 ln x x

and separate the variables to get:

  y - 1 2   d y =   4 ln x x   d x

And then we can integrate to get:

y 1 2 1 2 = ( 4 ) ( ln 2 x 2 ) + C
2 y = 2 ln 2 x + C

Using y(e)=1 we get:

2 = 2 ln 2 e + C
C = 0

Hence the particular solution is:

      2 y = 2 ln 2 x
y = ln 2 x
      y = ln 4 x

Validation:
1. x = e y = ln 4 e = 1 QED
2. d y d x = 4 ln 3 x x = 4 ln x x ln 2 x = 4 y ln x x QED

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