Linear systems of first order DE’s:
x’=A(t)x+g(t)
Initial conditions: x(t0)=x0
Existence/uniqueness theorem: The Initial Value Problem x’=A(t)x+g(t), x(t0)=x0 has a unique solution in the largest open interval I in t that contains t0 for which A(t) and g(t) are continuous.
General solution of homogeneous problem: x’=A(t)x
If we think of this as L[x]=x’-Ax=0 we see that L is linear in the usual sense of a linear operation. Hence, for the homogeneous case superposition of solutions applies
General solution on interval I where A is continuous:
If x’=Ax is an nxn system (n first order de’s) then the general solution can be expressed as a superposition of n solutions φ1, φ2,…, φn
x=c1 φ1+c2 φ2+…+cn φn provided that we can satisfy any set of initial conditions at some point with a solution of this form. This will be true if for some t=t0 in the interval I the system
c1φ1(t0)+c2 φ2(t0)+…+cn φn(t0)=x0
has a solution for any x0. In turn this is true if and only if det Ф(t0)≠0, where Ф(t) is the matrix whose columns are the solutions φ1, φ2,…, φn
If x=c1 φ1+c2 φ2+…+cn φn is the general solution then we call Ф(t) a fundamental matrix (of solutions) for the system x’=Ax.
Any solution of x’=Ax can then be written in the form x= c1 φ1+c2 φ2+…+cn φn= Ф(t)c where c is a vector of coefficients. Then the solution of the initial value problem x’=Ax, x(to)= x0 can be written x= Ф(t) Ф-1(t0) x0 The matrix Ф(t) Ф-1(t0) is a matrix where each column is a combination of the columns of Ф(t) and so Ф(t) Ф-1(t0) is a another fundamental matrix, Ψ(t), of the system x’=Ax with the property that Ψ(t0)=I and the solution of x’=Ax, x(to)= x0 is simply x= Ψ (t)x0 We say that Ψ is a fundamental matrix normalized at t=t0 if this holds.
Linear independence of solutions: We say solutions φ1, φ2,…, φn are linearly independent (on I) if no solution is a linear combination of the other solutions on the interval I. If φ1, φ2,…, φn is a linearly independent set of n solutions then Ф(t) is a fundamental matrix of solutions and det Ф(t) is nonzero for any t in the interval I. The reason this is true is because any set of solutions φ1, φ2,…, φn that fails to be a fundamental set must, for some t0 have the property that Ф(t0) is singular, and in turn there is a nonzero vector c of coefficients such that Ф(t0)c=0. Then the solution x= Ф(t)c satisfies zero initial conditions at t=t0 and hence must be identically zero. Now Ф(t)c≡0 means that one of the columns is, for all t in I, a linear combination of the other columns, i.e. the columns of Ф(t) are linearly dependent. We then argue that if the columns are independent, Ф(t) must be nonsingular at each t in I and hence a fundamental matrix.
Nonhomogeneous linear systems: x’=Ax+g
x=xh+xp is the general solution where the meaning is homogeneous general solution plus particular solution as in the scalar case.
There is a simple formula for xp (which I’ll refer to as the variation of parameters formula):
is the particular solution that satisfies xp(t0)=0. We can also write it in terms of an indefinite integral:
Constant coefficient case: x’=Ax+g(t) , A is constant
x’=Ax, homogeneous case, look for exponential solutions x=eλtv we find that this is a solution if and only if λ is an eigenvalue of A and v is a corresponding eigenvector.
Fundamental matrix: If A has a full set of real eigenvectors the fundamental matrix Ф can be written as
Fundamental matrix normalized at t=0
Fundamental matrix normalized at t0
General solution examples in real and complex cases
Matrix exponential and its properties
General solution when matrix is deficient
Transformational approach – decouple the system in the case of a full set of eigenvectors, use a similarity transformation to convert to Jordan for in the case that you don’t have a full set.
ψ-1(t)= ψ (-t) given normalized at t=0
exp(-At)exp(At)=exp(0t)=I
or simply note that ψ (-t)ψ (t)x0=x0 for any x0