Find a continuous function on $mathbb{R}$ such that the following initial value problem has two solutions












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$dot{x}=f(x)$ , $x(0)=0$ When I limit myself to functions that are continuous on all of the real numbers I can only think of function with 1 unique solution or infinite solutions. Could someone give me an example with two different solutions?






ordinary-differential-equations







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  • 1




    $begingroup$
    The standard example for the difference of Cauchy-Peano and Picard-Lindelöf is $dot x=2sqrt{|x|}$ or similar.
    $endgroup$
    – LutzL
    Jan 20 at 16:07
















0












$begingroup$


$dot{x}=f(x)$ , $x(0)=0$ When I limit myself to functions that are continuous on all of the real numbers I can only think of function with 1 unique solution or infinite solutions. Could someone give me an example with two different solutions?






ordinary-differential-equations







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    The standard example for the difference of Cauchy-Peano and Picard-Lindelöf is $dot x=2sqrt{|x|}$ or similar.
    $endgroup$
    – LutzL
    Jan 20 at 16:07














0












0








0





$begingroup$


$dot{x}=f(x)$ , $x(0)=0$ When I limit myself to functions that are continuous on all of the real numbers I can only think of function with 1 unique solution or infinite solutions. Could someone give me an example with two different solutions?






ordinary-differential-equations







share|cite|improve this question











$endgroup$




$dot{x}=f(x)$ , $x(0)=0$ When I limit myself to functions that are continuous on all of the real numbers I can only think of function with 1 unique solution or infinite solutions. Could someone give me an example with two different solutions?






ordinary-differential-equations




ordinary-differential-equations






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 20 at 15:08









Bernard

122k740116




122k740116










asked Jan 20 at 15:01









RogerRoger

847




847








  • 1




    $begingroup$
    The standard example for the difference of Cauchy-Peano and Picard-Lindelöf is $dot x=2sqrt{|x|}$ or similar.
    $endgroup$
    – LutzL
    Jan 20 at 16:07














  • 1




    $begingroup$
    The standard example for the difference of Cauchy-Peano and Picard-Lindelöf is $dot x=2sqrt{|x|}$ or similar.
    $endgroup$
    – LutzL
    Jan 20 at 16:07








1




1




$begingroup$
The standard example for the difference of Cauchy-Peano and Picard-Lindelöf is $dot x=2sqrt{|x|}$ or similar.
$endgroup$
– LutzL
Jan 20 at 16:07




$begingroup$
The standard example for the difference of Cauchy-Peano and Picard-Lindelöf is $dot x=2sqrt{|x|}$ or similar.
$endgroup$
– LutzL
Jan 20 at 16:07










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