given the differential equation: 𝑑2𝑦 𝑑𝑡2 + 12 𝑑𝑥 𝑑𝑡 + 15𝑥 = 35; 𝑡 ≥ 0 using matlab program, find: 𝑥(𝑡) when 𝑥(0) = 0 and 𝑥̇(0) = 1. in matlab

To solve the given differential equation in MATLAB, we can use the dsolve function or convert the equation to a system of first-order equations and use ode45. Here, we'll use dsolve. However, first, we need to correct and clarify the given differential equation. It seems there's a confusion in the variables used. Assuming the intent was to provide a second-order differential equation in terms of (x) and (t), and given the nature of the equation provided, let's interpret it as:

[ \frac{d^2x}{dt^2} + 12 \frac{dx}{dt} + 15x = 35 ]

With initial conditions (x(0) = 0) and (x'(0) = 1).

main.m
% Define the differential equation
syms x(t)
ode = diff(x,2) + 12*diff(x) + 15*x == 35;

% Define the initial conditions
ic = [x(0) == 0, diff(x,1)(0) == 1];

% Solve the differential equation
sol = dsolve(ode, ic);

% Display the solution
disp(sol);
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This code defines the differential equation using symbolic variables, specifies the initial conditions, and solves the equation using dsolve. The solution will be displayed in the command window.

To visualize the solution, you can use the ode45 function for numerical solution and plot the result:

main.m
% Convert the second-order differential equation into a system of first-order equations
% Let x1 = x, x2 = dx/dt
% Then dx1/dt = x2, dx2/dt = -12*x2 - 15*x1 + 35

% Define the model
model = @(t, x) [x(2); -12*x(2) - 15*x(1) + 35];

% Define the initial conditions and time span
x0 = [0; 1];  % x(0) = 0, x'(0) = 1
tspan = [0 10];  % Time span from 0 to 10

% Solve the differential equation numerically
[t, x] = ode45(model, tspan, x0);

% Plot the solution
plot(t, x(:,1));
xlabel('t');
ylabel('x(t)');
title('Solution of the differential equation');
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This numerical approach allows for a graphical representation of (x(t)) over time.