Proving single solution to an initial value problem y' = \left|\frac{1}{1+x^2}

iyiswad9k 2022-05-01 Answered

Proving single solution to an initial value problem
y'=|11+x2+sin|x2+arctany2||,      y(x0)=y0
or each (x0,y0)R×R I need to prove that there is a single solution defined on R

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Answers (1)

Zemmiq34
Answered 2022-05-02 Author has 11 answers

Step 1
Let
g(x,y)=sin|x2+arctan(y2)|=sin(x2+arctan(y2))
This function is derivable, hence
gy(x,y)=2yy4+1cos{(x2+arctan{(y2)})}
gy is continuous thus for each J there is a constant LJ so that
|g(x,y1)g(x,y2)|LJ|y1y2|
therefore
|f(x,y1)f(x,y2)|
=||11+x2+g(x,y1)||11+x2+g(x,y2)
|11+x2+g(x,y1)11+x2g(x,y2)|
=|g(x,y1)g(x,y2)|LJ|y1y2|
and f(x,y) is Lipschitz continuous on the 𝑦 variable, and this is true for each box J×(,) and therefore there is a single solution on R for each (x0,y0)R×R

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