#$&* course Mth 279 8/5 Query 22 Differential Equations*********************************************
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: Find the limit as t -> 0 of the matrix [ sin(t) / t, t cos(t), 3 / (t + 1); e^(3 t), sec(t), 2 t / (t^2 - 1) ] pictured as YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: take limit of each entry =|lim sin(t)/t , 0 , 3; 1, 1, 0| do L’H of sin(t)/t = cos(t)/1 =1 so lim A(t) = |1 ,0,3; 1,1,0| confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 2
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: Find A ' (t) and A ''(t), where the derivatives are with respect to t and the matrix is A = [ sin(t), 3 t; t^2 + 2, 5 ] pictured as YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: A = |sin(t), 3t; t^2+2, 5| A’(t) = |cos(t), 3; 2t, 0| A”(t) = |-sin(t), 0; 2,0| confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 2
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: Write the system of equations y_1 ' = t^2 y_1 + 3 y_2 + sec(t) y_2 ' = sin(t) y _1 + t y_2 - 5 in the form y ' = P(t) y + g(t), where P(t) is a 2 x 2 matrix and y and g(t) are 2 x 1 column vectors. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: y’ = [y’1; y’2] g(t) = [sec(t); -5] P(t) = [t^2, 3; sin(t),t] [y’1; y’2] = [t^2, 3; sin(t),t]*[y1;y2] + [sec(t); -5] confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 2
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: If A '' = [1, t; 0, 0] with A(0) = [ 1, 1; -2, 1] A(1) = [-1, 2; -2, 3 ] then what is the matrix A(t)? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 1) take integral twice of A” A(t) = [t^2/2 + C_11t+D_11, t^3/6+C_12t+D_12; C_21t+D_21, C_22t+D_22] A(0) = [1,1;-2,1] = [D_11, D_12; D_21, D_22] A(1) = [-1,2;-2,3] = [1/2+C_11(1)+1, 1/6+C_12(1)+1; C_21+(-2), C_22(1)+1] -1 = 1/2+ C_11+1; C_11 = -2.5 2= 1/6+C_12+1 ; C_12 = 5/6 -2 = C_21-2; C_21=0 3= C_22+1; C_22=2 A(t) = [ t^2/2 -2.5t+1, t^3/6+5/6t_1; -2, 2t+1] confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 2
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: Find the matrix A(t), defined by A(t) = integral( B(s) ds, s from 0 to t), where B = [ e^s, 6s; cos(2 pi s), sin(2 pi s) ]. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: A(t) = int(B(s))from 0 to t A(t) = [e^t - e^0, 3t^2-3(0)^2; 1/(2pi)*sin(2pit)-1/(2pi)*sin(2pi(0)), -1/(2pi)*(cos(2pi(t)-cos(2pi(0))) = [e^t-1, 3t^2; 1/(2pi)*sin(2pit), -1/(2pi)*(cos(2pi*t)-1)] confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 1
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating:" Self-critique (if necessary): ------------------------------------------------ Self-critique rating: Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!