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(Solved): Q. 6. Let us consider the Cauchy Problem (CP) x^(')=f(t,x),x(t_(0))=x_(0) where f(:)/(b)ar (D)->R ...
Q. 6. Let us consider the Cauchy Problem (CP)
x^(')=f(t,x),x(t_(0))=x_(0)where
f(:)/(b)ar (D)->R,D={(t,x)||t-t_(0)|<=a,|x-x_(0)|<=b}Peano says that a solution always exists in
|t-t_(0)|<=\alpha , where
\alpha =min{a,(b)/(M)}, and
M=max_((t,x)i(n)/(b)ar (D))|f(t,x)|, if
f(t,x)is continuous. (a) Give an example to show that this condition is sufficient but not necessary. (b) Show, by giving an example, that (CP) may still have a solution and this solution may be unique, however,
fdoes not satisfy Lipschtiz condition. Justify your answer.
