Proving single solution to an initial value problem

Tony Mccarthy

Tony Mccarthy

Answered question

2022-03-29

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

Answer & Explanation

Ashley Olson

Ashley Olson

Beginner2022-03-30Added 12 answers

Step 1
Let
g(x,y)=sinx2+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)LJy1-y2
therefore
f(x,y1)-f(x,y2)
=11+x2+g(x,y1)-11+x2+g(x,y2)
11+x2+g(x,y1)-11+x2-g(x,y2)
g(x,y1)-g(x,y2)LJy1-y2
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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