Solving a Linear System
Math 308
In this example, we see how to solve a linear system of the form x'=Ax.
Suppose
= A๎
๎
๎
๎๎
๎
๎
๎
1โ2
4โ5
and we want to solve x' = Ax,
where x = ๎
๎
๎
๎๎
๎
๎
๎
( )x t
( )y t
The differential equation written in matrix form above is really two first order equations:
x'(t) = x(t) - 2y(t)
y'(t) = 4x(t) - 5y(t)
We define the problem in Maple by entering these equations individually:
> eq1 := diff(x(t),t) = x(t)-2*y(t);
:= eq1 =
โ
โ
t( )x t โ ( )x t2 ( )y t
> eq2 := diff(y(t),t) = 4*x(t)-5*y(t);
:= eq2 =
โ
โ
t( )y t โ 4 ( )x t5 ( )y t
We can use the dsolve function to solve the system. As usual, the first argument to dsolve is the
problem to be solved, and the second argument holds the functions to be found. In this case, the
problem is a system, so we put both equations in brackets. Also, there are two functions to be found,
so we also put them in brackets.
> sol := dsolve([eq1,eq2],[x(t),y(t)]);
:= sol { }, = ( )y t + _C1 e( )โ3t_C2 e( )โt = ( )x t +
1
2_C1 e( )โ3t_C2 e( )โt
Note that Maple has returned the two functions in a list, enclosed in curly brackets. To access the
elements of the list, we use the notation sol[i], where i is an integer. It is important to also notice that
Maple did not put x(t) first; the order tends to be random. It might even be different the next time we
enter the exact same command.
> sol[2];
