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1. Our goal is to find a real-valued function u(x, t) solving 1 (1) ut +2ux = 0, 1+x² (This is called the transport equation.) We will do this in stages. (a) Suppose v = v(x, t) satisfies (2) Ut = 0, Compute v(x, t) for all x, t. u(x, 0) v(x,0) = - 1 1+x² (b) Suppose u satisfies the equations in (1) and define v(x, t) := u(x + 2t, t). Show that v satisfies the equations in (2). (c) Combine (a) and (b) to deduce an explicit function u satisfying (1).

1. Our goal is to find a real-valued function u(x, t) solving 1 (1) ut +2ux = 0, 1+x² (This is called the transport equation.) We will do this in stages. (a) Suppose v = v(x, t) satisfies (2) Ut = 0, Compute v(x, t) for all x, t. u(x, 0) v(x,0) = - 1 1+x² (b) Suppose u satisfies the equations in (1) and define v(x, t) := u(x + 2t, t). Show that v satisfies the equations in (2). (c) Combine (a) and (b) to deduce an explicit function u satisfying (1).

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter14: Discrete Dynamical Systems
Section14.3: Determining Stability
Problem 1YT
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1. Our goal is to find a real-valued function u(x, t) solving 1 (1) ut +2ux = 0, 1+x² (This is called the transport equation.) We will do this in stages. (a) Suppose v = v(x, t) satisfies (2) Ut = 0, Compute v(x, t) for all x, t. u(x, 0) v(x,0) = - 1 1 + x² (b) Suppose u satisfies the equations in (1) and define v(x, t) := u(x + 2t, t). Show that v satisfies the equations in (2). (c) Combine (a) and (b) to deduce an explicit function u satisfying (1).
Transcribed Image Text:1. Our goal is to find a real-valued function u(x, t) solving 1 (1) ut +2ux = 0, 1+x² (This is called the transport equation.) We will do this in stages. (a) Suppose v = v(x, t) satisfies (2) Ut = 0, Compute v(x, t) for all x, t. u(x, 0) v(x,0) = - 1 1 + x² (b) Suppose u satisfies the equations in (1) and define v(x, t) := u(x + 2t, t). Show that v satisfies the equations in (2). (c) Combine (a) and (b) to deduce an explicit function u satisfying (1).
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