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Differential Equations Lecture Work Solutions 244

Differential Equations Lecture Work Solutions 244 - 9 Show...

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9. Show that the initial value problem u t + u x = x u ( x, x ) = 1 has no solution. Give a reason for the problem. Note that the initial line in this case is NOT the x axis ( t = 0) but the line t = x . The ODEs are dx dt = 1 , du dt = x and the initial condition for each x ( t = x ) = ξ, u ( x ( t = x ) , t = x ) = 1 Solve the characteristic equation x = t + K and use the initial condition ξ = ξ + K so K = 0 and the characteristic is x = t This is the only line, there is no family as in other problems. Substitute in the
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