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fufufu
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2nd order DE..."largest interval" confusion
Determine the largest t-interval on which therem 3.1 guarantees the existence of a unique solution:
y'' + 3t^2y' + 2y = sin(t) ...y(1) = 1 ...y'(1) = -1
theroem 3.1 is the one that states if p(t) and q(t) and g(t) are continuous functions on the interval (a,b) then the IVP y'' + p(t)y' + q(t)y = g(t) has unique solution defined on the entire interval , (a,b) .
if i therefore locate any discontinuities on each function in the equation, then that will determine what the intervals are.
p(t) = 3t^2 and g(t) sin(t) and q(t) = 2(*) are all continuous, so the largest interval is -infin<t<infin, which is the answer in back of book.
But what was the point of the initial values then?
(my geuss) --do i look at the tvalues and consider which is the largest interval that that t-value falls in? (in this case i only have one interval, so it makes it easy, but am i explaining it correctly?)
(*) my other question is: in this problem, is q(t) = 2? if there is no t in the term, then how can i ask if there are any discontinuities there?
thanks for any help in understanding this..
Homework Statement
Determine the largest t-interval on which therem 3.1 guarantees the existence of a unique solution:
y'' + 3t^2y' + 2y = sin(t) ...y(1) = 1 ...y'(1) = -1
Homework Equations
theroem 3.1 is the one that states if p(t) and q(t) and g(t) are continuous functions on the interval (a,b) then the IVP y'' + p(t)y' + q(t)y = g(t) has unique solution defined on the entire interval , (a,b) .
The Attempt at a Solution
if i therefore locate any discontinuities on each function in the equation, then that will determine what the intervals are.
p(t) = 3t^2 and g(t) sin(t) and q(t) = 2(*) are all continuous, so the largest interval is -infin<t<infin, which is the answer in back of book.
But what was the point of the initial values then?
(my geuss) --do i look at the tvalues and consider which is the largest interval that that t-value falls in? (in this case i only have one interval, so it makes it easy, but am i explaining it correctly?)
(*) my other question is: in this problem, is q(t) = 2? if there is no t in the term, then how can i ask if there are any discontinuities there?
thanks for any help in understanding this..