Differential Equations Lecture Work Solutions 261

Differential Equations Lecture Work Solutions 261 - 9. t +...

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9. ρ t + ρu x + x =0 u t + uu x =0 with u ( x, 0) = f ( x )and ρ ( x, 0) = g ( x ). The set of ODEs is dx dt = u du dt =0 Solve the second ODE to get u ( x ( t ) ,t )= K = f ( ξ ) . Now susbstitute this in the Frst ODE and solve x = f ( ξ ) | {z } = u t + ξ So ξ = x tu and u ( x, t )= f ( x tu ) Now use this u in the Frst PDE ρ t + ρu x + x =0 The ODEs for this are dx dt = u, u is known! and dt = ρu x Solve the Frst to get
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