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 this system of linear first-order differential equations.

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

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) = diff(u(t), t) == 3*u(t) + 4*v(t) diff(v(t), t) == 3*v(t) - 4*u(t)

Solve the system using the `dsolve`

function
which returns the solutions as elements of a structure.

S = dsolve(odes)

S = struct with fields: v: [1×1 sym] u: [1×1 sym]

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 vSol(t) = S.v

uSol(t) = C2*cos(4*t)*exp(3*t) + C1*sin(4*t)*exp(3*t) vSol(t) = C1*cos(4*t)*exp(3*t) - C2*sin(4*t)*exp(3*t)

Alternatively, store `u(t)`

and `v(t)`

directly
by providing multiple output arguments.

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

uSol(t) = C2*cos(4*t)*exp(3*t) + C1*sin(4*t)*exp(3*t) vSol(t) = C1*cos(4*t)*exp(3*t) - C2*sin(4*t)*exp(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)*exp(3*t) vSol(t) = cos(4*t)*exp(3*t)

Visualize the solution using `fplot`

.

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

Solve differential equations in matrix form by using `dsolve`

.

Consider this system of differential equations.

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

The matrix form of the system is

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

Let

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

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) = diff(x(t), t) == x(t) + 2*y(t) + 1 diff(y(t), t) == t - x(t) + y(t)

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

xSol(t) = (2*t)/3 + 2^(1/2)*C2*exp(t)*cos(2^(1/2)*t) + 2^(1/2)*C1*exp(t)*sin(2^(1/2)*t) + 1/9 ySol(t) = C1*exp(t)*cos(2^(1/2)*t) - t/3 - C2*exp(t)*sin(2^(1/2)*t) - 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) = (2*t)/3 + (17*exp(t)*cos(2^(1/2)*t))/9 - (7*2^(1/2)*exp(t)*sin(2^(1/2)*t))/9 + 1/9 ySol(t) = - t/3 - (7*exp(t)*cos(2^(1/2)*t))/9 - (17*2^(1/2)*exp(t)*sin(2^(1/2)*t))/18 - 2/9

Visualize the solution using `fplot`

.

clf fplot(ySol) hold on fplot(xSol) grid on legend('ySol','xSol','Location','best')