Analytical solution of ODEs

Hi everyone, I have a question regarding the analytical solution of the ODE`s. I have an ODE like "dx/dt = -2*x" with x = 8 at t = 0 and I have to solve it using Euler Method and Analytically on Matlab. Luckily, my friend got the solution of the code which is given as below. I am clear about Euler solution but can anyone explain that how he has solved it analytically? It will be a lot of help to me!

So if I need to solve other ODEs, How I can solve it? Like dx/dt = -4/x with x = 8 at t = 0.

clear all close all

h=1; % dxdt = -2*x; x(1) = 1;

t=[0:h:8];

for i=1:length(t)-1

x(i+1) = x(i) + -2*x(i)*h;

end

texact = [0:0.02:8];

xexact = exp(-2.*texact);

% size(x)

% size(t)

plot(t,x)

hold on

plot(texact,xexact,'k-.')

legend('numerical','analytical')

[Torsten]

The code is for the ODE

dx/dt = -2*x
x(0) = 1

Best wishes

Torsten.

Answer 1

[Star Strider]

It’s a simple separable first-degree differential equation:

dx/dt = -2*x
dx/x = -2*dt              —>  integrate both sides to get:
log(x(t)) = -2*t
x(t) = exp(-2*t)

Q.E.D.

[Torsten]

x(t) = 8*exp(-2*t)

Best wishes

Torsten.

[Star Strider]

Correct.

I forgot the initial condition.

Still waking up here (UCT-6).

Answer 2

[alex mloso]

A =