How to solve Coupled Differential Equations

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Tomasz
Tomasz 2021-5-9
评论： Walter Roberson 2025-2-8，18:22
Hi
how to solve that set of equations
y'-x'=2-x
2x'-y'=3+2y

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回答（3 个）

Star Strider
Star Strider 2021-5-9
Ran in:
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) = 
.
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Walter Roberson
Walter Roberson 2025-2-8，18:22
Ran in:
Of course, for given x0 and y0, it is easy enough to plot.
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: exp(-t)*(y0 - exp(t)/2 + 1/2) x: exp(t)*(x0 - y0/2 + (9*exp(-t))/4 - 9/4) + (exp(-t)*(y0 - exp(t)/2 + 1/2))/2
x(t) = simplify(S.x, 500)
x(t) = 
y(t) = simplify(S.y, 500)
y(t) = 
X0 = -2;
Y0 = 1.5;
sX = subs(x, [x0, y0], [X0, Y0])
sX(t) = 
sY = subs(y, [x0, y0], [X0, Y0])
sY(t) = 
fplot(sX, [0 5])
fplot(sY, [0 5])

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Walter Roberson
Walter Roberson 2021-5-9
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
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Walter Roberson
Walter Roberson 2021-5-9
Ran in:
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
Walter Roberson 2021-5-9
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

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Sam Chak
Sam Chak 2025-2-8，9:08
Ran in:
Hi @Lorenzo
Primarily, coupled ordinary differential equations (ODEs) in additive form can be decoupled using the substitution and elimination method (Grade 10 algebra). In this simple system, we can decouple the ODEs simultaneously using the matrix method.
The coupled ODEs
can be expressed in matrix form
and simplified to
.
Because of the square matrix property , we can manipulate the matrix equation to become
so that we can solve the following decoupled ODEs
.
Method 1: Numerical approach
% Describe the ODEs inside the function
function dx = ode(t, x)
dx(1) = 5 - 1*x(1) + 2*x(2); % ODE 1
dx(2) = 7 - 2*x(1) + 2*x(2); % ODE 2
dx = [dx(1)
dx(2)]; % arranged in column vector
end
% Call ode45 to numerically solve the system
tspan = [0 10]; % time interval from 0 to 10
xinit = [1; 0]; % initial condition at time 0
[t, x] = ode45(@ode, tspan, xinit);
plot(t, x), grid on, xlabel('Time'), legend('x(t)', 'y(t)')
title('Numerical Solutions')
●
Method 2: Symbolical approach to obtain the analytical solutions (using @Star Strider's code):
%% Requires Symbolic Math Toolbox
syms x(t) y(t) x0 y0
Dx = diff(x);
Dy = diff(y);
ode1 = Dy - Dx == 2 - x
ode1(t) = 
ode2 = 2*Dx - Dy == 3 + 2*y
ode2(t) = 
S = dsolve(ode1, ode2, x(0)==1, y(0)==0) % specify the initial condition here
S = struct with fields:
y: - exp(t/2)*sin((7^(1/2)*t)/2)*((exp(-t/2)*(21*sin((7^(1/2)*t)/2) + 25*7^(1/2)*cos((7^(1/2)*t)/2)))/14 - (17*7^(1/2))/14) - exp(t/2)*cos((7^(1/2)*t)/2)*((exp(-t/2)*(21*co... x: - ((3*exp(t/2)*cos((7^(1/2)*t)/2))/4 + (7^(1/2)*exp(t/2)*sin((7^(1/2)*t)/2))/4)*((exp(-t/2)*(21*cos((7^(1/2)*t)/2) - 25*7^(1/2)*sin((7^(1/2)*t)/2)))/14 - 3/2) - ((3*exp(...
x(t) = simplify(S.x, 500)
x(t) = 
y(t) = simplify(S.y, 500)
y(t) = 
fplot(x(t), [0 10]), hold on, grid on, ylim([-300, 500])
fplot(y(t), [0 10]), hold off, xlabel('Time'), legend('x(t)', 'y(t)')
title('Analytical Solutions')

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