If the function f(x,y) is continuous near the point (a,b), then at least one solution of the differential equation y' = f(x,y) exists on some open interval I containing the point x = a and, moreover, that if in addition the partial of derivative - is continuous near (a,b) then this solution is unique on some (perhaps smaller) interval J. Determine whether existence of at least one solution of the given initial value problem thereby guaranteed and, if so, dy whether uniqueness of that solution is guaranteed. dy =x° -y°: y(7) = 2 dx Select the correct choice below and fill in the answer box(es) to complete your choice. (Type an ordered pair.) O A. The theorem implies the existence of at least one solution because f(x,y) is continuous near the point of This solution is unique because ду is also continuous near that same point. %3D Ов. of The theorem implies the existence of at least one solution because f(x,y) is continuous near the point. However, this solution is not necessarily unique because is not continuous near that same point. dy O C. The theorem does not imply the existence of at least one solution because f(x,y) is not continuous near the point
If the function f(x,y) is continuous near the point (a,b), then at least one solution of the differential equation y' = f(x,y) exists on some open interval I containing the point x = a and, moreover, that if in addition the partial of derivative - is continuous near (a,b) then this solution is unique on some (perhaps smaller) interval J. Determine whether existence of at least one solution of the given initial value problem thereby guaranteed and, if so, dy whether uniqueness of that solution is guaranteed. dy =x° -y°: y(7) = 2 dx Select the correct choice below and fill in the answer box(es) to complete your choice. (Type an ordered pair.) O A. The theorem implies the existence of at least one solution because f(x,y) is continuous near the point of This solution is unique because ду is also continuous near that same point. %3D Ов. of The theorem implies the existence of at least one solution because f(x,y) is continuous near the point. However, this solution is not necessarily unique because is not continuous near that same point. dy O C. The theorem does not imply the existence of at least one solution because f(x,y) is not continuous near the point
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:If the function f(x,y) is continuous near the point (a,b), then at least one solution of the differential equation y' = f(x,y) exists on some open interval I containing the point x = a and, moreover, that if in addition the partial
of
is continuous near (a,b) then this solution is unique on some (perhaps smaller) interval J. Determine whether existence of at least one solution of the given initial value problem is thereby guaranteed and, if so,
ду
derivative
whether uniqueness of that solution is guaranteed.
dy
=x° - y°; y(7) =2
dx
3
%3D
Select the correct choice below and fill in the answer box(es) to complete your choice.
(Type an ordered pair.)
А.
df
This solution is unique because
ду
The theorem implies the existence of at least one solution because f(x,y) is continuous near the point
is also continuous near that same point.
О В.
The theorem implies the existence of at least one solution because f(x,y) is continuous near the point
of
However, this solution is not necessarily unique because
ду
is not continuous near that same point.
O C. The theorem does not imply the existence of at least one solution because f(x,y) is not continuous near the point
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