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A more detailed version of Theorem 1 says that, if the function f(x, y) is continuous near the point (a, b), then at least one so- lution 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 derivative af/ay is continuous near (a, b), then this solution is unique on some (perhaps smaller) interval J. In Problem determine whether ex- istence of at least one solution of the grven initial value prob- lem is thereby guaranteed and, if so, whether uniqueness of that solution is guaranteed. dy = In(1+ y²); y(0) = 0 dx

A more detailed version of Theorem 1 says that, if the function f(x, y) is continuous near the point (a, b), then at least one so- lution 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 derivative af/ay is continuous near (a, b), then this solution is unique on some (perhaps smaller) interval J. In Problem determine whether ex- istence of at least one solution of the grven initial value prob- lem is thereby guaranteed and, if so, whether uniqueness of that solution is guaranteed. dy = In(1+ y²); y(0) = 0 dx

Advanced Engineering Mathematics
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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A more detailed version of Theorem 1 says that, if the function f(x, y) is continuous near the point (a, b), then at least one so- lution 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 derivative af/ay is continuous near (a, b), then this solution is unique on some (perhaps smaller) interval J. In Problem determine whether ex- istence of at least one solution of the grven initial value prob- lem is thereby guaranteed and, if so, whether uniqueness of that solution is guaranteed. dy = In(1+ y²); y(0) = 0 dx
Transcribed Image Text:A more detailed version of Theorem 1 says that, if the function f(x, y) is continuous near the point (a, b), then at least one so- lution 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 derivative af/ay is continuous near (a, b), then this solution is unique on some (perhaps smaller) interval J. In Problem determine whether ex- istence of at least one solution of the grven initial value prob- lem is thereby guaranteed and, if so, whether uniqueness of that solution is guaranteed. dy = In(1+ y²); y(0) = 0 dx
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