How to solve Coupled Differential Equations

Hi

how to solve that set of equations

y'-x'=2-x

2x'-y'=3+2y

Answer 1

[Star Strider]

Try this —

syms x(t) y(t) x0 y0 
Dx = diff(x);
Dy = diff(y);
ode1 = Dy-Dx == 2 - x

ode1(t) = 

ode2 = 2-Dy - Dy == 3 + 2*y

ode2(t) = 

S = dsolve(ode1, ode2, x(0)==x0, y(0)==y0)

S = struct with fields:
    y: [1×1 sym]
    x: [1×1 sym]

x(t) = simplify(S.x, 500)

x(t) = 

y(t) = simplify(S.y, 500)

y(t) = 

.

Answer 2

[Walter Roberson]

yprime - xprime == 2-x, 2*xprime - yprime == 3 + 2*y
2*yprime - 2*xprime == 4-2*x, 2*xprime - yprime == 3 + 2*y
2*yprime - 2*xprime + 2*xprime - yprime == 4-2*x + 3 + 2*y
yprime == 7 - 2*x + 2*y
7 - 2*x + 2*y - xprime == 2*x
7 - 2*x + 2*y - 2*x == xprime
xprime = 7 - 4*x + 2*y

So...

function dxy = odefun(t, xy)
   dxy = [7 - 4*xy(1) + 2*xy(2); 7 - 2*xy(1) + 2*xy(2)];
end

[Tomasz]

this:

2*yprime - 2*xprime == 4-2*x, 2*xprime - yprime == 3 + 2*y

2*yprime - 2*xprime + 2*xprime - yprime == 4-2*x + 3 + 2*y

is it OK??

[Walter Roberson]

2*yprime - 2*xprime == 4-2*x, 2*xprime - yprime == 3 + 2*y

add the two equations

(2*yprime - 2*xprime) + (2*xprime - yprime) == (4-2*x) + (3 + 2*y)

[Walter Roberson]

syms xprime yprime x y
eqn = [yprime - xprime == 2-x, 2*xprime - yprime == 3 + 2*y]

eqn = 

sol = solve(eqn, [xprime, yprime])

sol = struct with fields:
    xprime: [1×1 sym]
    yprime: [1×1 sym]

sol.xprime

ans = 

sol.yprime

ans = 

Hmmm, where did I go wrong?

[Walter Roberson]

I did make a mistake, but it was the step after you indicated. I copied as 2*x instead of 2-x . The corrected version is

yprime - xprime == 2-x, 2*xprime - yprime == 3 + 2*y
2*yprime - 2*xprime == 4-2*x, 2*xprime - yprime == 3 + 2*y
2*yprime - 2*xprime + 2*xprime - yprime == 4-2*x + 3 + 2*y
yprime == 7 - 2*x + 2*y
7 - 2*x + 2*y - xprime == 2 - x   %corrected
7 - 2*x + 2*y - (2 - x) == xprime
xprime = 5 - x + 2*y

So...

function dxy = odefun(t, xy)
   dxy = [5 - xy(1) + 2*xy(2); 7 - 2*xy(1) + 2*xy(2)];
end