# hw3 - Math 425/525 Spring 2011 Jerry L Kazdan Problem Set 3...

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Math 425/525, Spring 2011 Jerry L. Kazdan Problem Set 3 DUE: In class Thursday, Feb. 10 Late papers will be accepted until 1:00 PM Friday . 1. Solve u x + u y + u = e x + 2 y with u ( x , 0 ) = 0. 2. Find the general solution of u xy = x 2 y for the function u ( x , y ) . 3. Find the general solution of the inhomogeneous equation u tt - u xx = 1 + 2 x for the function u ( x , t ) , where - < x < (an inﬁnite string). 4. Solve the wave equation (for an inﬁnite string) u tt = c 2 u xx with initial conditions u ( x , 0 ) = ln ( 1 + x 2 ) and u t ( x , 0 ) = 4 + x . 5. [Weinberger, p.17 #3] A string of length L = 1 with ﬁxed end points is initially ﬁxed in the position u ( x , 0 ) = sin π x and is released at time t = 0 (so its initial velocity is zero). Find its subsequent motion. 6. [THE DULCIMER] Solve the wave equation u tt = c 2 u xx with initial conditions u ( x , 0 ) = 0 and u t ( x , 0 ) = g ( x ) , where g ( x ) = 1 if | x | < a and g ( x ) = 0 for | x | ≥ a . This corresponds to hitting

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hw3 - Math 425/525 Spring 2011 Jerry L Kazdan Problem Set 3...

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