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Unformatted text preview: Math 425, Spring 2011 Jerry L. Kazdan Problem Set 7 DUE: Thursday March 24 [ Late papers will be accepted until 1:00 PM Friday ]. 1. Suppose u is a twice differentiable function on R which satisfies the ordinary differential equation u 00 + b ( x ) u- c ( x ) u = , where b ( x ) and c ( x ) are continuous functions on R with c ( x ) > 0 for every x ∈ ( , 1 ) . a) Show that u cannot have a positive local maximum in the interval ( , 1 ) , that is, have a local maximum at a point p where u ( p ) > 0. Also show that u cannot have a negative local minimum in ( , 1 ) . [The example u 00 + u = 0 has u ( x ) = sin x as a solution, which does have posivive local maxima and negative local minima. This shows that some assumption, such as our c ( x ) > is needed.] b) If u ( ) = u ( 1 ) = 0, prove that u ( x ) = 0 for every x ∈ [ , 1 ] . c) If u satisfies 4 u xx + 3 u yy- 5 u = in a region D ⊂ R 2 , show that it cannot have a local positive maximum. Also show that u cannot have a local negative minimum.cannot have a local negative minimum....
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