Transform the given differential equation or system into an equivalent system of first-order differential equations. X" = 8x - y + 6z, y"' = x+ y - 3z, z" = 5x-y - z . . . * 1 ' = x 2 , y 1 ' = y 2 , 2 1 ' = 2 2 X'= y2Three 140gal fermentation vats are connected as indicated in the figure, and the mixtures in each tank are kept uniform by stirring. Denote by xi(t) the amount (in pounds) of alcohol in tank Ti at time t (i = 1, 2, 3). Suppose that the mixture circulates between the tanks at the rate of 14 gal/min. Derive the equations. , 10x1 = -x1 +x3 10x2'= x1-x2 , 10x3 = x2-x3 Calculate the concentration of alcohol in each tank. The alcohol concentration in tank T1 is ':| The alcohol concentration in tank T2 is The alcohol concentration in tank T3 is gallons pound ' gallons. pounds gallon ' pounds. \fFind a particular solution of the indicated linear system that satisfies the initial conditions x1(0)= 3. x2(0) = - 7. 9 -4 1 2 x'= x;x1=.:5t ,::(2=e'5t 14 '9 1 7 c:> The particular solution is )(,l (t) = D and x2(t) = D. First verify that the given vectors are solutions of the given system. Then use the Wronskian to show that they are linearly independent. Finally, write the general solution of the system. 12 16 1 3 1 1 x'= 10 14 1 x;x1=e'2' -2 .x2=e3t -1 .x3=e4t -1 1o 10 3 2 1 0 -2t 12 -16 1 12 16 -1 3e 10 14 1x1= 1o 14 1 - 2e-2t =D=x1' 10 1o 3 1o 10 3 2e- 21