HINT GIVEN:

1. (10 points) Solve each of the following differential equations using Laplace transformation technique. Evaluate the obtained x(t) and give a general sketch of x(t) versus time (no need to use a calculator). Verify your solution by using MATLAB where you are asked to create three plots: 1) plot of the closed-form solution of x(t), 2) numerical solution of x(t), and plot of X(s) using MATLAB impulse command. 4 plots must agree. (see hint) (a) * +48 + 3x = 0 x(0) = 0, (0) = 2 + 4x = 0 x(0) = 5, 8(0)=0 (b) MATLAB %ME3017 solution of linear equation exercise %PLOT 1: analytical solution of y tt=0:0.1:2; figure (1) plot (tt, 1-2*exp(-4*tt) +exp(-8*tt)) grid %PLOT 2: Numerical solution of y(t) figure (3) syms y(t) Dy=diff(y) ode=diff(y,t,2)==32 -12 * diff(y,t,1)-32*y condl = y(0) == 0 cond2 = Dy (0) == 0 conds = [condi cond2] ySol (t) = dsolve (ode, conds) plot (tt, subs (ySol, tt)) grid %PLOT 3: MATLAB impulse plot of Y(s) figure (2) Y=tf(32, [1 12 32 01) impulse (Y, tt) grid Figure Figure 2 Edit View insert Tools Desktop Window Help Impulse Response Amplitude 0 02 04 06 08 12 14 16 0 14 16 18 2 0 02 04 06 08 12 14 16 18 02 04 05 08 12 Time (seconds) All agree with the general sketch 1. (10 points) Solve each of the following differential equations using Laplace transformation technique. Evaluate the obtained x(t) and give a general sketch of x(t) versus time (no need to use a calculator). Verify your solution by using MATLAB where you are asked to create three plots: 1) plot of the closed-form solution of x(t), 2) numerical solution of x(t), and plot of X(s) using MATLAB impulse command. 4 plots must agree. (see hint) (a) * +48 + 3x = 0 x(0) = 0, (0) = 2 + 4x = 0 x(0) = 5, 8(0)=0 (b) MATLAB %ME3017 solution of linear equation exercise %PLOT 1: analytical solution of y tt=0:0.1:2; figure (1) plot (tt, 1-2*exp(-4*tt) +exp(-8*tt)) grid %PLOT 2: Numerical solution of y(t) figure (3) syms y(t) Dy=diff(y) ode=diff(y,t,2)==32 -12 * diff(y,t,1)-32*y condl = y(0) == 0 cond2 = Dy (0) == 0 conds = [condi cond2] ySol (t) = dsolve (ode, conds) plot (tt, subs (ySol, tt)) grid %PLOT 3: MATLAB impulse plot of Y(s) figure (2) Y=tf(32, [1 12 32 01) impulse (Y, tt) grid Figure Figure 2 Edit View insert Tools Desktop Window Help Impulse Response Amplitude 0 02 04 06 08 12 14 16 0 14 16 18 2 0 02 04 06 08 12 14 16 18 02 04 05 08 12 Time (seconds) All agree with the general sketch