Determine whether the initial value problem y′ = cos(t + y), y(t_0) = y_0 has a unique solution defined on all of R and show F(t,y) is lipchitz in y on any S
Determine whether the initial value problem y′ = cos(t + y), y(t_0) = y_0 has a unique solution defined on all of R and show F(t,y) is lipchitz in y on any S
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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Determine whether the initial value problem y′ = cos(t + y), y(t_0) = y_0 has a unique solution defined on all of R and show F(t,y) is lipchitz in y on any S
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Step 1
Suppose an initial value problem is provided. If the function is continuous on a rectangular domain that includes the initial value, the equation exhibits a solution. The equation exhibits a unique solution in that very rectangular domain if and are continuous on that domain. It is referred to as the existence and uniqueness theorem. The continuity is also verified by determining whether it is a Lipschitz function. When a function is Lipschitz, it is uniformly continuous and hence continuous.
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