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verifying definition interval of given differential equation
Hi everyone,
I need your help to check the method I followed to solve this given differential equation:
I cant actually write powers in LaTex, how do you do that?
[tex] { y' = (x3)(y^(2)2 ), y(3) = 6 } [/tex]
now, supposing [tex] y != +/ 1 [/tex] is it possible to rewrite the first eq. as:
[tex] int[3,y] dy/(y^(2) 1) = int[6,x] (x3) dx [/tex]
hence:
[tex] (1/2)ln(1+y)/(1y)  (1/2)ln(2) = x^(2)/2 3x [/tex]
some counts and we obtain:
[tex] y = (2(e^(x^(2)6x)))/(12(e^(x^(2)6x))) [/tex]
At this point we look at the conditions imposed during the process, namely [tex] y != +/ 1 [/tex]:
per [tex]y != 1[/tex], we found that, being:
[tex] (2(e^(x^(2)6x)))/(12(e^(x^(2)6x))) != 1 [/tex]
it should be true that:
[tex] e^(x^(2)6x) = 1/4 [/tex]
therefore this condition does not create problems.
It follows [tex] y != 1 [/tex] and it should not be true that:
[tex] e^(x^(2)6x) = 1/4 [/tex]
which equation is solved for [tex] x = 3 +/ sqrt(9 ln4) [/tex].
Highlighted these details, we have a look at the denominator, who requires the condition:
[tex] e^(x^(2)6x) != 1/2 [/tex]
in order to determine the local interval of definiton.
This equation leads to exclude [tex] x [/tex] values equal to [tex] x = 3 +/ sqrt(9  ln2) [/tex].
All these considerations take to write this kind of graph :
hence we deduct that the (max?) largest definition interval in which the solution is defined is:
[tex] x belongs to ( 3  sqrt(9  ln2) , 3 + sqrt(9 ln4) ) [/tex]
Is everything correct?
Thanks!
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