# hw2 - k so the equations of motion are mu 00 1 = k u 2-u 1...

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Math 425/525, Spring 2011 Jerry L. Kazdan Problem Set 2 DUE: In class Thursday, Feb. 3 Late papers will be accepted until 1:00 PM Friday . 1. Find the solution U ( t ) : = ( u 1 ( t ) , u 2 ( t )) of u 0 1 = u 1 u 0 2 = u 1 - u 2 with the initial conditions U ( 0 ) = ( u 1 ( 0 ) , u 2 ( 0 )) = ( 10 ) . 2. Find the solution U ( t ) of u 0 1 = 2 u 1 - u 2 u 0 2 = 3 u 1 - 2 u 2 with U ( 0 ) = ( 0 , 1 ) . 3. By only a slight modiﬁcation of your solution of the previous problem, ﬁnd the general solution of u 00 1 = 2 u 1 - u 2 u 00 2 = 3 u 1 - 2 u 2 4. One simple model of a diatomic molecule is of masses m and M connected by a spring with spring constant
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Unformatted text preview: k so the equations of motion are mu 00 1 = k ( u 2-u 1 ) Mu 00 2 = k ( u 1-u 2 ) Solve these equations by ﬁnding the normal modes of oscillation. Interpret your results with a diagram. 5. a) Find the solution u ( x , y ) of u x + 3 u y-u = 0 with u ( , y ) = 1 + y 2 . b) Find the solution v ( x , y ) of v x + 3 v y-v = 1 with v ( , y ) = 1 + y 2 . [Last revised: January 29, 2011] 1...
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