Proving single solution to an initial value problemy'=11+x2+sin|x2+arctany2|,&nbsp;&nbsp;&nbsp;(x0)=y0or each&nbsp;(x0,y0)∈R×R&nbsp;I need to prove that there is a single solution defined on&nbsp;R

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Proving single solution to an initial value problemy&#039;=11+x2+sin|x2+arctany2|,&amp;nbsp;&amp;nbsp;&amp;nbsp;(x0)=y0or each&amp;nbsp;(x0,y0)∈R×R&amp;nbsp;I need to prove that there is a single solution defined on&amp;nbsp;R

Ashley Olson · Accepted Answer

Step 1Letg(x,y)=sinx2+arctan(y2)=sin(x2+arctan(y2))This function is derivable, hencegy(x,y)=2yy4+1cos(x2+arctan(y2))gy&amp;nbsp;is continuous thus for each J there is a constant&amp;nbsp;LJ&amp;nbsp;so thatg(x,y1)-g(x,y2)≤LJy1-y2thereforef(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-y2and&amp;nbsp;f(x,y)&amp;nbsp;is Lipschitz continuous on the 𝑦 variable, and this is true for each box&amp;nbsp;J×(−∞,∞)&amp;nbsp;and therefore there is a single solution on&amp;nbsp;R&amp;nbsp;for each&amp;nbsp;(x0,y0)∈R×R

Proving single solution to an initial value problem

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