Solve a System of Differential Equations

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Solve a system of several ordinary differential equations in several variables by using the dsolve function, with or without initial conditions. To solve a single differential equation, see Solve Differential Equation.

Solve System of Differential Equations

Solve this system of linear first-order differential equations.

$\begin{array}{l} \frac{du}{dt} = 3 u + 4 v, \\ \frac{dv}{dt} = - 4 u + 3 v . \end{array}$

First, represent $u$ and $v$ by using syms to create the symbolic functions u(t) and v(t).

syms u(t) v(t)

Define the equations using == and represent differentiation using the diff function.

ode1 = diff(u) == 3*u + 4*v;
ode2 = diff(v) == -4*u + 3*v;
odes = [ode1; ode2]

odes(t) = 
 $(\begin{array}{c} \frac{\partial}{\partial t} u (t) = 3 u (t) + 4 v (t) \\ \frac{\partial}{\partial t} v (t) = 3 v (t) - 4 u (t) \end{array})$

Solve the system using the dsolve function which returns the solutions as elements of a structure.

S = dsolve(odes)

S = struct with fields:
    v: C1*cos(4*t)*exp(3*t) - C2*sin(4*t)*exp(3*t)
    u: C2*cos(4*t)*exp(3*t) + C1*sin(4*t)*exp(3*t)

If dsolve cannot solve your equation, then try solving the equation numerically. See Solve a Second-Order Differential Equation Numerically.

To access u(t) and v(t), index into the structure S.

uSol(t) = S.u

uSol(t) =  $C_{2} \cos (4 t) e^{3 t} + C_{1} \sin (4 t) e^{3 t}$

vSol(t) = S.v

vSol(t) =  $C_{1} \cos (4 t) e^{3 t} - C_{2} \sin (4 t) e^{3 t}$

Alternatively, store u(t) and v(t) directly by providing multiple output arguments.

[uSol(t),vSol(t)] = dsolve(odes)

uSol(t) =  $C_{2} \cos (4 t) e^{3 t} + C_{1} \sin (4 t) e^{3 t}$

vSol(t) =  $C_{1} \cos (4 t) e^{3 t} - C_{2} \sin (4 t) e^{3 t}$

The constants C1 and C2 appear because no conditions are specified. Solve the system with the initial conditions u(0) == 0 and v(0) == 0. The dsolve function finds values for the constants that satisfy these conditions.

cond1 = u(0) == 0;
cond2 = v(0) == 1;
conds = [cond1; cond2];
[uSol(t),vSol(t)] = dsolve(odes,conds)

uSol(t) =  $\sin (4 t) e^{3 t}$

vSol(t) =  $\cos (4 t) e^{3 t}$

Visualize the solution using fplot.

fplot(uSol)
hold on
fplot(vSol)
grid on
legend("uSol","vSol",Location="best")

Figure contains an axes object. The axes object contains 2 objects of type functionline. These objects represent uSol, vSol.

Solve Differential Equations in Matrix Form

Solve differential equations in matrix form by using dsolve.

Consider this system of differential equations.

$\begin{array}{l} \frac{dx}{dt} = x + 2 y + 1, \\ \frac{dy}{dt} = - x + y + t . \end{array}$

The matrix form of the system is

$[\begin{array}{c} x^{'} \\ y^{'} \end{array}] = [\begin{array}{c} 1 & 2 \\ - 1 & 1 \end{array}] [\begin{array}{c} x \\ y \end{array}] + [\begin{array}{c} 1 \\ t \end{array}] .$

Let

$Y = [\begin{array}{c} x \\ y \end{array}], A = [\begin{array}{c} 1 & 2 \\ - 1 & 1 \end{array}], B = [\begin{array}{c} 1 \\ t \end{array}] .$

The system is now $Y^{'} = A Y + B .$ .

Define these matrices and the matrix equation.

syms x(t) y(t)
A = [1 2; -1 1];
B = [1; t];
Y = [x; y];
odes = diff(Y) == A*Y + B

odes(t) = 
 $(\begin{array}{c} \frac{\partial}{\partial t} x (t) = x (t) + 2 y (t) + 1 \\ \frac{\partial}{\partial t} y (t) = t - x (t) + y (t) \end{array})$

Solve the matrix equation using dsolve. Simplify the solution by using the simplify function.

[xSol(t),ySol(t)] = dsolve(odes);
xSol(t) = simplify(xSol(t))

xSol(t) = 
 $\frac{2 t}{3} + \sqrt{2} C_{2} e^{t} \cos (\sqrt{2} t) + \sqrt{2} C_{1} e^{t} \sin (\sqrt{2} t) + \frac{1}{9}$

ySol(t) = simplify(ySol(t))

ySol(t) = 
 $C_{1} e^{t} \cos (\sqrt{2} t) - \frac{t}{3} - C_{2} e^{t} \sin (\sqrt{2} t) - \frac{2}{9}$

The constants C1 and C2 appear because no conditions are specified.

Solve the system with the initial conditions $u (0) = 2$ and $v (0) = - 1$ . When specifying equations in matrix form, you must specify initial conditions in matrix form too. dsolve finds values for the constants that satisfy these conditions.

C = Y(0) == [2;-1];
[xSol(t),ySol(t)] = dsolve(odes,C)

xSol(t) = 
 $\begin{array}{l} \sqrt{2} e^{t} σ_{2} (\frac{17 \sqrt{2}}{18} + \frac{e^{- t} (4 σ_{1} + \sqrt{2} σ_{2} + 6 t σ_{1} + 6 \sqrt{2} t σ_{2})}{18}) - \sqrt{2} e^{t} σ_{1} (\frac{e^{- t} (4 σ_{2} - \sqrt{2} σ_{1} + 6 t σ_{2} - 6 \sqrt{2} t σ_{1})}{18} + \frac{7}{9}) \\ where \\ σ_{1} = \sin (\sqrt{2} t) \\ σ_{2} = \cos (\sqrt{2} t) \end{array}$

ySol(t) = 
 $\begin{array}{l} - e^{t} σ_{1} (\frac{17 \sqrt{2}}{18} + \frac{e^{- t} (4 σ_{1} + \sqrt{2} σ_{2} + 6 t σ_{1} + 6 \sqrt{2} t σ_{2})}{18}) - e^{t} σ_{2} (\frac{e^{- t} (4 σ_{2} - \sqrt{2} σ_{1} + 6 t σ_{2} - 6 \sqrt{2} t σ_{1})}{18} + \frac{7}{9}) \\ where \\ σ_{1} = \sin (\sqrt{2} t) \\ σ_{2} = \cos (\sqrt{2} t) \end{array}$

Visualize the solution using fplot.

figure
fplot(ySol)
hold on
fplot(xSol)
grid on
legend("ySol","xSol",Location="best")

Figure contains an axes object. The axes object contains 2 objects of type functionline. These objects represent ySol, xSol.

Solve a System of Differential Equations

Solve System of Differential Equations

Solve Differential Equations in Matrix Form

See Also