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# quiz9_sol - ~ v = y ~ i x ~ j using 2 steps with a time...

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Date: March 30, 2011 MA3160-08 Quiz 9 No Calculators! Justify all answers! Name (print): Solutions 1. (5pts.) Find the flow line (in parametrized form) of the vector field ~ F = ~ i + 3 x ~ j through the point (1, 1) at time t = 0. Solution The flow lines are defined by the vector di ff erential equation ~ r 0 ( t ) = ~ F ( ~ r ( t )), which leads to the two di ff erential equations ( x 0 ( t ) = 1 y 0 ( t ) = 3 p x ( t ) . Integration of the first equation gives x ( t ) = t + x 0 , where x 0 is the constant of integration. x 0 is determined from the initial condition x (0) = 1, which implies x 0 = 1 and, therefore, x ( t ) = t + 1. Substitution of x ( t ) = t + 1 into the second equation gives y 0 ( t ) = 3 t + 1, which, after integration yields y ( t ) = 2( t + 1) 3 / 2 + y 0 . Again, y 0 is determined from the second initial condition y (0) = 1, which implies that y 0 = - 1 and, therefore, y ( t ) = 2( t + 1) 3 / 2 - 1. It follows that the parametric form of the flow line through the point (1,1) is given by x ( t ) = t + 1 y ( t ) = 2( t + 1) 3 / 2 - 1 Note that in the xy -plane, this is the curve y = 2 x 3 / 2 - 1. 2. (5 pts.) Use Euler’s method to approximate the flow through (1, 2) of the vector field
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Unformatted text preview: ~ v = y ~ i + x ~ j using 2 steps with a time interval of Δ t = . 1. Solution Euler’s method is given by ~ r n + 1 = ~ r n + Δ t ~ F ( ~ r n ) , which in component form is ( x n + 1 = x n + Δ tF 1 ( x n , y n ) y n + 1 = y n + Δ tF 2 ( x n , y n ) . Using Δ t = . 1, F 1 ( x n , y n ) = y n and F 2 ( x n , y n ) = x n yields the equations ( x n + 1 = x n + . 1 y n y n + 1 = y n + . 1 x n . Therefore, ( x 1 = x + . 1 y = 1 + . 1 · 2 = 1 . 2 y 1 = y + . 1 x = 2 + . 1 · 1 = 2 . 1 and ( x 2 = x 1 + . 1 y 1 = 1 . 2 + . 1 · 2 . 1 = 1 . 41 y 2 = y 1 + . 1 x 1 = 2 . 1 + . 1 · 1 . 2 = 2 . 22 . Therefore, the ﬂow through (1, 2) of the vector ﬁeld ~ v = y ~ i + x ~ j at t = . 2 is approximately x (0 . 2) ≈ x 2 = 1 . 41 y (0 . 2) ≈ y 2 = 2 . 22...
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