Please Use Matlab to solve it. a. Use the command dsolve to solve the differenti

Question

Please Use Matlab to solve it.

a. Use the command dsolve to solve the differential equations below, each with initial conditions y(0) = 1 and y'(0) = 0. i. y" + 3y = 0 ii. y" + 4y' + 29y = 0 iii. y" -y/36 = 0 iv. y" + 2y' + y = 0 v. y" + 6y' + 6y = 0 b. Graph each of the solutions in a. in the same window with 0 lessthanorequalto t lessthanorequalto 10. c. Which of these equations does not represent a mechanical vibration? Why not? In your comments, explain how to recognize that the equation cannot describe a mechanical vibration i. from the graph of the solution, and also ii. from the coefficients of the original equation. (Recall how we have interpreted the coefficients m, b, and k where my" + by' + ky = 0.) d. In your comments, classify the other four solutions as undamped, underdamped, critically damped, or overdamped.

Explanation / Answer

clc;
clear;

y=dsolve('Dy^2=-3*y','y(0)=1')
y=dsolve('Dy^2+4*Dy=-29*y','y(0)=1')
y=dsolve('Dy^2=(1/36)*y','y(0)=1')
y=dsolve('Dy^2+2*Dy=-y','y(0)=1')
y=dsolve('Dy^2+6*Dy=-6*y','y(0)=1')

result:

y =

-(2*i + 3^(1/2)*t)^2/4
-(2*i - 3^(1/2)*t)^2/4


y =

4/29 - (exp((5*i)/2 - (29*t)/4 + log(5*i - 2) - 1)/exp(wrightOmega((5*i)/2 - (29*t)/4 + log((5*i)/2 - 1) - 1)) + 2)^2/29
4/29 - (exp(log(- 5*i - 2) - (29*t)/4 - (5*i)/2 - 1)/exp(wrightOmega(log(- (5*i)/2 - 1) - (29*t)/4 - (5*i)/2 - 1)) + 2)^2/29
4/29 - (exp((5*i)/2 - (29*t)/4 + log(2 - 5*i) - 1)/exp(wrightOmega((5*i)/2 - (29*t)/4 + log(1 - (5*i)/2) + pi*i - 1)) - 2)^2/29


y =

(t/6 + 2)^2/4
(t/6 - 2)^2/4


y =

1 - (exp(-wrightOmega(i*pi - 1/2*t - 1))*exp(- 1/2*t - 1) - 1)^2


y =

solve(3^(1/2)*log(2*(9 - 6*y)^(1/2) + 6) - (3 - 2*y)^(1/2) = 3^(1/2)*log(2*3^(1/2) + 6) - 3^(1/2)*t - 1, y)
solve(- 3^(1/2)*log(6 - 2*(9 - 6*y)^(1/2)) - (3 - 2*y)^(1/2) = 3^(1/2)*t - 3^(1/2)*log(6 - 2*3^(1/2)) - 1, y)

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