**Unformatted text preview: **w ( t ) = 0 for all t ≥ 0. c) [Uniqueness] Say the functions u ( t ) and v ( t ) both satisfy the same equation (1) and also u ( ) = v ( ) and u ( ) = v ( ) . Show that u ( t ) = v ( t ) for all t ≥ 0. 4. Say u ( x , y ) has the property that ∂ u ∂ y = 0 for all points ( x , y ) and that u ( x , ) = sin3 x . Find u ( x , y ) . What if instead u satisﬁes ∂ u ∂ y = 2 xy ? 5. A function u ( x , y ) satisﬁes u x + 3 u y =0. Find a change of variables x = as + bt y = cs + dt so that in the new ( s , t ) variables u satisﬁes ∂ u ∂ s = 0. [Last revised: January 15, 2011] 1...

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