How can I solve this differential equation? : x y d x -...

vegetatzz8s

vegetatzz8s

Answered

2022-11-25

How can I solve this differential equation? : x y   d x - ( x 2 + 1 )   d y = 0

Answer & Explanation

SquingonottokQqk

SquingonottokQqk

Expert

2022-11-26Added 8 answers

we have in differential form:
x y   d x - ( x 2 + 1 )   d y = 0
If we put in standard form, and collect terms:
1 y y   d y d x = x x 2 + 1
Which is a First Order Separable Ordinary Differential Equation, so we can separate the variables to get:
  1 y   d y =   x x 2 + 1   d x
We can manipulate the RHS integral as follows:
  1 y   d y = 1 2     2 x x 2 + 1   d x
And now both integrals are standard results, so integrating give us:
ln | y | = 1 2 ln | x 2 + 1 | + C
Noting that we require areal solution, and writing C = ln A , we get:
ln y = ln A x 2 + 1
Giving us the General Solution:
y = A x 2 + 1

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