1. MATLAB PROGRAM 1 : Matrix to Linear system of Equation

% 1) Matrix to Linear system of Equation
% Take matrix A from the user
A = input('Enter your matrix A of order m * n: ','s');
A = str2num(A); % Convert the string input to a numeric matrix

% Get the number of rows and columns of matrix A
[m, n] = size(A); % assign values to m & n

% Prompt user to enter Matrix B
B = input('Enter your matrix B of order m * 1: ','s');
B = str2num(B); % Convert the string input to a numeric matrix

% Check if the number of rows in A matches the number of rows in B
if size(B, 1) ~= m
error('Error: The number of rows in Matrix B must match the number of rows in Matrix A.');
end

% Initialize symbolic variables
vars = sym('x', [1, n]); % Creates symbolic variables x1, x2, ..., xn
X = vars.'; % Transpose symbolic variable array to a column vector

% Define the system of linear equations
equations = A * X == B;

% Solve the system of equations
solution = solve(equations, X);

% Display the solution
disp('The solution to the system of equations is:')
disp(solution)

disp('The system of equations is set up successfully.');

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MATLAB PROGRAM 2 : System of Linear Equation to Matrix

% Equation to Matrix
% Prompt user for the number of variables
num_vars = input('Enter the number of variables: ');

% Initialize symbolic variables
vars = sym('x', [1, num_vars]);

% Prompt user for the number of equations
num_eqns = input('Enter the number of equations: ');

% Inform the user about the symbolic variables
fprintf('Kindly enter equations using the following variables: ');
disp(vars);

% Initialize an empty symbolic array to hold equations
equations = sym([]);

% Loop to input equations
for k = 1:num_eqns
    % Ask for the equation in string format
    eqStr = input(sprintf('Enter equation %d (e.g., x1 + 2*x2 + x3 = 5): ', k), "s");
    % Append the equation to the symbolic array
    equations(end+1) = str2sym(eqStr);
end

% Convert the equations to matrix form [A|B]
[A, B] = equationsToMatrix(equations, vars);

% Display the results
disp('Matrix A (coefficients):');
disp(A);
disp('Matrix B (constants):');
disp(B);

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MATLAB PROGRAM 3 : Unique, Infinite & No solution of Linear System

A = input('Enter your matrix A of order m*n: ');
B = input('Enter your column matrix B of order m*1: ');

% Ensure B is a column vector and its size matches A's rows
[m, n] = size(A);
if size(B, 1) ~= m || size(B, 2) ~= 1
    error('Matrix B must have dimensions m*1, where m matches the number of rows in A.');
end

% Construct augmented matrix
A_B = [A B]; % Augmented matrix

% Calculate ranks
r1 = rank(A_B); % Rank of augmented matrix
r2 = rank(A);   % Rank of coefficient matrix

% Determine nature of the solution
if r1 == r2 && r1 == n
    disp('Given system is consistent and has a unique solution');
elseif r1 == r2 && r1 < n
    disp('Given system is consistent and has infinite solutions');
else
    disp('Given system is inconsistent and has no solution');
end

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MATLAB PROGRAM 4 : Unique, Infinite & No solution of Linear System

syms x y z
equations = [1*x + 2*y + 3*z == 3, 3*x + 6*y + 9*z == 9, 7*x + 8*y + 10*z == 34];

% Solve the system of equations
sol = solve(equations, [x, y, z], 'ReturnConditions', true);

% Extract solutions and conditions
x = sol.x;
y = sol.y;
z = sol.z;
conditions = sol.conditions;

% Display the solutions
disp('Solution for x:');
disp(x);
disp('Solution for y:');
disp(y);
disp('Solution for z:');
disp(z);

% Display any conditions
disp('Conditions for the solution:

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MATLAB PROGRAM 5 : Linearly Dependent & Independent Vectors

num_vect = input('Enter the number of vectors: ');

% Initialize an empty matrix to hold vectors
vect_mat = [];

for k = 1:num_vect
    vector = input(sprintf('Enter your %d vector as a row vector: ', k));
    
    % Validate that all vectors have the same dimension
    if ~isempty(vect_mat) && length(vector) ~= size(vect_mat, 2)
        error('All vectors must have the same dimension.');
    end
    
    vect_mat = [vect_mat; vector]; % Append the vector row-wise
end

% Transpose the matrix to have vectors as columns
A = vect_mat';

% Check the rank of the matrix
if rank(A) == num_vect
    disp('Vectors are linearly independent (LI)');
else
    disp('Vectors are linearly dependent (LD)');
end

--------------------
INPUT:
Enter the number of vectors: 3
Enter your 1 vector as a row vector: [1, 2, 3]
Enter your 2 vector as a row vector: [4, 5, 6]
Enter your 3 vector as a row vector: [7, 8, 9]
--------------------


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MATLAB PROGRAM 6 : Eigen Values & Eigen Vectors

A = input('Enter the square matrix A: ');

% Check if A is a square matrix
[m, n] = size(A);
if m ~= n
    error('Your matrix is not a square matrix.');
end

% Compute eigenvalues and eigenvectors
[V, D] = eig(A); % V is the eigenvector matrix, D is the diagonal matrix of eigenvalues

% Display eigenvalues and corresponding eigenvectors
for k = 1:m
    fprintf('The %d eigenvalue is: %f\n', k, D(k, k));
    fprintf('The corresponding eigenvector is:\n');
    disp(V(:, k));
end

--------------------
INPUT:
Enter the square matrix A: [2, 0; 0, 3]

--------------------

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MATLAB PROGRAM 7 : Limit / Indeterminate Forms

% Symbolic Variable
syms x 

% Example Limit Calculations
fprintf('Example Calculations:\n');
fprintf('Limit of cos(x) as x -> pi/2 (radians): %f\n', limit(cos(x), x, pi/2));
fprintf('Limit of sin(x) as x -> 90 degrees: %f\n', limit(sin(x), x, pi/2)); % Fixed 'sind' to 'sin' and angle in radians

% Interactive Programming
g = input("Enter the function g(x) as a string (e.g., 'x^2 + 3*x - 5'): ", 's');
g = str2sym(g); % Convert the input string to a symbolic expression
a = input("Enter the value 'a' for which limit x tends to 'a': ");

% Calculate the Limit
h = limit(g, x, a);

% Display the Result
fprintf("The limit of the function g(x) = %s as x tends to %g is: %s\n", char(g), a, char(h));


--------------------
INPUT:
Enter the function g(x) as a string (e.g., 'x^2 + 3*x - 5'): x^2 + 2*x + 1
Enter the value 'a' for which limit x tends to 'a': -1

--------------------

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MATLAB PROGRAM 8 : Taylor's and Maclaurin's Theorem

% 8) Taylor's and Maclaurin's Theorem

% Syntax Examples
disp('Syntax Examples:');
disp(taylor(exp(x), x, 2)); % Taylor expansion around x = 2
disp(taylor(exp(x), x, 0)); % Maclaurin's series (a = 0)
disp(taylor(exp(x), x, 0, 'Order', 10)); % Expand function up to degree 9

% Sympref Settings
% Uncomment to control polynomial display style
% sympref("PolynomialDisplayStyle", "ascend"); % Expand series in ascending powers
% sympref("PolynomialDisplayStyle", "descend"); % Expand series in descending powers
% sympref("PolynomialDisplayStyle", "default"); % Expand series in default setting

% Expansion of Function with Multiple Variables
syms x y z
disp(taylor(sin(y) + cos(z), y, 0)); % Expand in powers of y
disp(taylor(sin(y) + cos(z), z, 0)); % Expand in powers of z
disp(taylor(sin(y) + cos(z), [y, z], 0)); % Expand in powers of y and z

% Interactive Programming
f = input("Enter the function f(x): ", 's');
a = input("Expansion at point a: ");
b = input("Order of the expansion is b: ");

% Convert function to symbolic expression
f = str2sym(f);

% Taylor Expansion
g = taylor(f, x, a, 'Order', b);

% Display the Result
fprintf("The expansion of f(x) = %s at x = %i is:\n%s\n", char(f), a, char(g));

-----------------------------------
INPUT
Enter the function f(x): x^2 + 3*x + 1
Expansion at point a: 1
Order of the expansion is b: 5
------------------------------------


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MATLAB PROGRAM 9 : Successive Differentiation

% Successive Differentiation
% Syntax
syms x y % Symbolic variables
diff(x^3 - 2*x, x, 2) % Differentiate function x^3 - 2*x 2 times w.r.t. x
diff(x^2*y - 3*x*y^3, x, 1) % First-order partial derivative w.r.t. x
diff(x^2*y - 3*x*y^3, y, 2) % Second-order partial derivative w.r.t. y

% Interactive Programming
syms x % Define symbolic variable x for interactive programming
f = input("Enter the function f(x): ", 's'); % Input as string
f = str2sym(f); % Convert input string to symbolic expression
n = input("Enter the order of the derivative n: ");

for i = 1:n
    f_d = diff(f, x, i); % Successive differentiation
    fprintf("The %i derivative of the function f(x) is %s \n", i, char(f_d));
end


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INPUT

Enter the function f(x): x^3 - 2*x

Enter the order of the derivative n: 2

-------------------

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MATLAB PROGRAM 10 : Successive Differentiation Using Partial

% 10) Successive Differentiation Using Partial Fraction
syms x
f = input('Enter your function f(x): ', 's'); % Input function as a string
f = str2sym(f); % Convert input string to symbolic expression

% Perform partial fraction decomposition
f_pf = partfrac(f); % Decompose into partial fractions
fprintf('The partial fraction of the function f(x) is: %s\n', char(f_pf));

% Input the required order of derivative
n = input('Enter the required order of derivative n: ');

for i = 1:n
    f_d = diff(f_pf, x, i); % Successive differentiation
    fprintf('The %i derivative of the function f(x) is: %s\n', i, char(f_d));
end


-------------------
INPUT

Enter your function f(x): (x^2 + 2*x + 1)/(x^2 - x)

Enter the required order of derivative n: 2


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MATLAB PROGRAM 11 : Exact &Non Exact Differential Equation

% 11) Exact & Non-Exact Differential Equation
syms x y C
% Define M(x, y) and N(x, y)
M = input('Enter M(x, y): ', 's'); % Input as string
N = input('Enter N(x, y): ', 's'); % Input as string
M = str2sym(M); % Convert to symbolic
N = str2sym(N); % Convert to symbolic

% Check for exactness
dM_dy = diff(M, y); % Partial derivative of M with respect to y
dN_dx = diff(N, x); % Partial derivative of N with respect to x
disp(['dM/dy = ', char(dM_dy)]);
disp(['dN/dx = ', char(dN_dx)]);

if dM_dy ~= dN_dx
    disp('Given differential equation is Non-Exact.');
    % Input Integrating Factor
    IF = input('Enter integrating factor mu(x, y): ', 's'); % Input as string
    IF = str2sym(IF); % Convert to symbolic
    
    % Modify M and N with the integrating factor
    M = M * IF;
    N = N * IF;
    
    % Recalculate after applying the integrating factor
    dM_dy = diff(M, y);
    dN_dx = diff(N, x);
    disp('Modified functions:');
    disp(['M(x, y): ', char(M)]);
    disp(['N(x, y): ', char(N)]);
    
    if dM_dy == dN_dx
        disp('The modified differential equation is now Exact.');
    else
        disp('Error: Even with the integrating factor, the equation is not Exact.');
        return;
    end
end

disp('Given differential equation is Exact.');
% Solve the Exact Differential Equation
M_intx = int(M, x); % Integrate M with respect to x
N_inty = int(N - diff(M_intx, y), y); % Integrate remaining terms with respect to y
GS = M_intx + N_inty; % General solution

% Display the General Solution
fprintf('The General Solution (GS) is: %s = C\n', char(GS));

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INPUT

Enter M(x, y): y*exp(x)
Enter N(x, y): exp(x)

OR

Enter M(x, y): 3*x^2*y
Enter N(x, y): x^3 + y



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MATLAB PROGRAM 12: Heat Flow

% 12) Heat Flow
syms T(x) x
% Input parameters
x1 = input('Enter the inner radius of the pipe (x1): ');
x2 = input('Enter the outer radius of the pipe (x2): ');
T1 = input('Enter the inner temperature of the pipe (T1): ');
T2 = input('Enter the outer temperature of the pipe (T2): ');
k = input('Enter coefficient of thermal conductivity (k): ');

% Calculate heat flux/loss (q)
q = -2 * pi * k * (T2 - T1) / log(x2 / x1);

% Display the heat flux
fprintf('Heat flux (q) = %.4f\n', q);

% Input the distance normal to the surface
a = input('Enter distance from the center (a): ');

% Validate that 'a' lies within the bounds of the pipe
if a < x1 || a > x2
    error('Distance "a" must lie between the inner and outer radius of the pipe (x1 <= a <= x2).');
end

% Calculate temperature at distance 'a'
Temp = ((T2 - T1) / log(x2 / x1)) * log(a / x1) + T1;

% Display the temperature
fprintf('Temperature at distance a (Temp) = %.4f\n', Temp);

----------------------------
INPUT

Enter the inner radius of the pipe (x1): 1
Enter the outer radius of the pipe (x2): 2
Enter the inner temperature of the pipe (T1): 300
Enter the outer temperature of the pipe (T2): 400
Enter coefficient of thermal conductivity (k): 50
Enter distance from the center (a): 1.5
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MATLAB PROGRAM 13 : Solving higher order LDE

% 13) Solving Homogeneous Linear Differential Equation (LDE)

% Define symbolic variables and differential equation
syms x y(x)

% Example predefined ODE
ode = diff(y, x, 2) + 5*diff(y, x) + 6*y == 0;

% Solve and expand the solution for the example ODE
sol = dsolve(ode);
disp('Solution for the predefined ODE:');
disp(expand(sol));

% Interactive Programming
disp('Interactive Mode: Solve your own ODE');
ode = input('Enter the Differential Equation (use y(x) for dependent variable): ', 's');
ode = str2sym(ode); % Convert input string to symbolic expression

% Solve and display the expanded solution
sol = dsolve(ode);
disp('Solution for the entered ODE:');
disp(expand(sol));


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INPUT

Enter the Differential Equation (use y(x) for dependent variable): diff(y,x,2) + 7*diff(y,x) + 10*y == 0

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% 14) Solving Non-Homogeneous Linear Differential Equation (LDE)

% Define symbolic variables and example differential equation
syms x y(x)

% Predefined Example ODE
ode = diff(y, x, 2) + 3*diff(y, x) + 2*y == exp(exp(x));

% Solve and expand the solution for the example ODE
sol = dsolve(ode);
disp('Solution for the predefined ODE:');
disp(expand(sol));

% Interactive Programming
disp('Interactive Mode: Solve your own ODE');
ode = input('Enter the Differential Equation (use y(x) for dependent variable): ', 's');
ode = str2sym(ode); % Convert input string to symbolic expression

% Solve and display the expanded solution
sol = dsolve(ode);
disp('Solution for the entered ODE:');
disp(expand(sol));

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INPUT

Enter the Differential Equation (use y(x) for dependent variable): diff(y,x,2) + 4*diff(y,x) + 3*y == sin(x)


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% 15) Application on Electric Circuit

% Step I: Generate differential equation from the given electric circuit
% Define symbolic variables
syms t q(t)

% Define the ODE based on the given circuit
% d^2q/dt^2 + 3*dq/dt + 2*q = sin(t)
ode = diff(q, t, 2) + 3*diff(q, t) + 2*q == sin(t);

% Display the ODE
disp('Differential Equation for the Electric Circuit:');
disp(ode);

% Step II: Solve the ODE using dsolve (higher-order LDE solver)
sol = dsolve(ode);

% Expand the solution for better readability
expanded_sol = expand(sol);

% Display the solution
disp('Solution for the given ODE:');
disp(expanded_sol);