30.3. HOMOGENEOUS PARTICULAR AND GENERAL SOLUTIONS 595

are linearly independent and this shows that the dimension of the solution space is n if A isn×n. In the above example, the dimension of the general solution is 4 because A is 4×4.

Now consider the general solution to

x′ = Ax+f

There is a very easy way to describe this. It is just the general solution to x′ = Ax added toxp where xp is any particular solution to the above nonhomogeneous equation.

Theorem 30.3.4 The general solution to x′=Ax+f consists of all solutions to this equa-tion. It is of the form Φ(t)c+xp where xp is a particular solution meaning x′p =Axp+f .

Proof: Anything of the form Φ(t)c+xp is a solution to x′ = Ax+f . It remains toverify that this is the only way it can happen. Let z′ = Az+f and consider (z−xp) .Then

(z−xp)′ = z′−x′p = Az+f− (Axp +f) = A(z−xp)

and so z−xp is a solution to x′=Ax. Therefore, from Theorem 30.3.2, there exists c∈Fn

such that z (t)=Φ (t)c+xp (t). ■

Example 30.3.5 Find the general solution to

x′ = Ax+f

where

A =

 2 −4 −23 −4 −2−3 10 6

 , f (t) =

 et sin te−t cos t

t

First I will find the fundamental matrix using the following syntax.>>syms s t; b=eye(3);

c=[2 -4 -2;3 -4 -2;-3 10 6];f=[exp(t)*sin(t);exp(-t)*cos(t);t];simplify(ilaplace(inv(s*b-c)))simplify(ilaplace(inv(s*b-c)*laplace(f)))The first line starting with “simplify” will give the fundamental matrix and the second

will give a particular solution. The claim about the first was already considered. As to thesecond, if x is a particular solution with zero initial condition,

sX (s) = AX (s)+F (s)

In the above syntax, the F (s) comes from laplace(f). Then

X (s) = (sI−A)−1 F (s)

and this involves inv(s*b-c)*laplace(f) in the above syntax. Then you do ilaplace to thisthing to get a particular solution. Try it. You will get a horrendous mess but Matlab has noproblem in doing it.

This has shown how to solve first order systems at least up to a suitable variation ofconstants formula. There is one other topic which is sometimes useful and that is theconvolution integral and its relation to the Laplace transform.