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Finding region of xy plane for which differential equation has a unique solution
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September 27th, 2013, 05:59
find_the_fun
Finding region of xy plane for which differential equation has a unique solution
Determine a region of the xy-plane for which the given differential equation would have a unique solution
whose graph passes through a point (x0 , y0 ) in the region.
x
dy
=y
dx
What does an xy-plane have to do with anything? I looked up the definition of unique solutions and here it is
Let R be a rectangular region in the xy-planed defined by a <=x<=b, c<=y<=d that contains the
point
(x0 , y0 ) in its interior. If f(x,y) and
∂df
∂dy
are continuous on R then there exists some interval
I0 : (x0 − h, x0 + h), h > 0 contained in [a/b] and a unique function y(x) defined on I0
that is a
solution of the initial value problem.
That's a bit difficult to digest. How do I proceed?
Fernando Revilla
September 27th, 2013, 13:57
Re: Finding region of xy plane for which differential equation has a unique solution
D ≡ x > 0 and D′ ≡ x < 0 the differential equation is equivalent to
∂f
y
1
y′ = f (x, y) = , and in both regions, f and
= are continuous, so and according to a well known
x
∂y
x
theorem, D and D′ are solutions to your question.
In each of the regions
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