Question

Problem 11: a. Use the Laplace approach in order to obtain the analytical solutions x(t) for the following Differential Equations using MATLAB (don't use DSOLVE): , Diff. Eqn. 1: * + 2x + 10x - Diff. Eqn. 2: 2x + 7* + 3x -0, *(0) - 0, X(0) - 3, *(0) - 0 4(0)-0

Answer 1

clc;
clear;
% Defining symbolic variables
syms x(t) s P Q
% Defining derivatives to use in equations
Dx=diff(x,t); % First derivative
D2x=diff(x,t,2); % Second derivative
eqn1=D2x+2*Dx+10*x==exp(-t); % 1st equation
eqn2=2*D2x+7*Dx+3*x==0; % 2nd equation
a=laplace(eqn1);% Finding laplace
b=laplace(eqn2);% Finding laplace
eqnlt1=subs(a,laplace(x(t),t,s),P);% Replacing laplace with Variable P
eqnlt2=subs(b,laplace(x(t),t,s),Q);% Replacing laplace with Variable Q
% Substuting the conditions x(0)=0;Dx(0)=0;
eqnlt1=subs(eqnlt1,[x(0),subs(diff(x(t), t), t, 0)],[0,0]);
% Substuting the conditions x(0)=3;Dx(0)=0;
eqnlt2=subs(eqnlt2,[x(0),subs(diff(x(t), t), t, 0)],[3,0]);
% Solving for X(s)
a2=solve(eqnlt1,P);
b2=solve(eqnlt2,Q);
% Finding the x(t) using Inverse laplace
a3=ilaplace(a2)
b3=ilaplace(b2)