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s1_95c - ACM 95/100c April 2 2004 Problem Set I 1 Solve the...

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ACM 95/100c April 2, 2004 Problem Set I 1.- Solve the first-order PDE ∂u ∂x + 2 ∂u ∂y = 0 with boundary conditions u = y 2 on x = 0, -∞ < y < . 2.- (a) Find the solution u of the equation ∂u ∂x + ∂u ∂y = 1 satisfying u = x 2 , on y = α x , α = 1. (b) Discuss existence and uniqueness in the case α = 1 considering, in particular, the initial condition u = x on y = x . 3. The density of traffic on a freeway (cars per unit length) is described by the function ρ ( x, t ), where x is the distance down traffic from an obstruction and t is the time. For a particular type of obstruction to traffic flow, the traffic density function ρ ( x, t ) is described by the first-order PDE ∂ρ ∂t + c ∂ρ ∂x = exp( - 3 x ) , where c is a constant. Obtain the general solution of this PDE with initial condition ρ ( x, 0) = f ( x ), where f ( x ) is given. 4.- i) Solve the initial value problems x 2 u x + y 2 u y = u 2 u = 1 on the initial curve y = 2 x and xu x + uu y = y u (1 , y ) = 3 y. ii) Consider the problem yu x - xu y = 0 u ( x, 0) = u 0 ( x ) .
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Show that if u 0 ( x ) = x 2
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