clc;

clear;

% Declare symbolic variables

syms t x y z

% Input the function

f = input('Enter the function f(x,y,z): ');

% Input variable relations

x_t = input('Enter x as a function of t: ');

y_t = input('Enter y as a function of t: ');

z_t = input('Enter z as a function of t: ');

% Partial derivatives

fx = diff(f, x);

fy = diff(f, y);

fz = diff(f, z);

% Total derivative using chain rule

dfdt = subs(fx, x, x_t)*diff(x_t, t) + ...

subs(fy, y, y_t)*diff(y_t, t) + ...

subs(fz, z, z_t)*diff(z_t, t);

% Display output

disp('Total derivative df/dt = ');

disp(simplify(dfdt));

2

clc;

clear;

syms y(x)

% Enter Bernoulli equation

eqn = input('Enter the Bernoulli equation: ');

% Solve the equation

sol = dsolve(eqn);

% Display solution

disp('Solution is:');

disp(sol);

3

clc;

clear;

% Input the function

f = input('Enter the function f(x,y): ');

% Input initial conditions

x0 = input('Enter initial value of x: ');

y0 = input('Enter initial value of y: ');

h = input('Enter step size h: ');

xn = input('Enter final value of x: ');

x = x0;

y = y0;

while x < xn

k1 = h * f(x, y);

k2 = h * f(x + h/2, y + k1/2);

k3 = h * f(x + h/2, y + k2/2);

k4 = h * f(x + h, y + k3);

y = y + (k1 + 2*k2 + 2*k3 + k4)/6;

x = x + h;

end

disp('The solution using RK-4 method is:');

disp(y);