The solution of differential equations has been the heart and soul of the physical sciences since the time of Isaac Newton. The question of whether or not a solution actually exists for an equation involving a derivative is vital. Fortunately for linear equations constructive proofs of the existence of solutions are readily available. One goal of the analysis is to be able to say that for a system of n linear equations there exists n independent solutions.

Preliminaries

For a square matrix M define its expontential function exp(M) which maps square matrices into square matrices of the same dimension as

exp(M) = I + M + M²/2! + M³/3! + …

where I is the identity matrix of the same dimensions as M. Questions of convergence will be dealt with later.

For a matrix multiplied by a scalar t the definition reduces to

exp(Mt) = I + Mt + M²t²/2! + M³t³/3! + …

Systems of Linear Homogeneous
Differential Equations and Their Solution

A set of linear differential equations of the form

dx₁/dt = a_1,1x_i + a_1,2x₂ + … + a_1,nx_n
dx₂/dt = a_2,1x_i + a_2,2x₂ + … + a_2,nx_n
......................................................
dx_n/dt = a_n,1x_i + a_n,2x₂ + … + a_n,nx_n

can be expressed as

dX/dt = AX(t)

where X(t) is an n dimensional column vector and A is the n×n matrix of the coefficients a_i,j. The solution requires the initial conditions X(t)=X₀ be satisfied.

Now consider the exponential matrix function

exp(At) = I + At + A²t²/2! + A³t³/3! + …

The derivative of the right-hand side (RHS) of the above with respect to t gives

d(exp(At))/dt = A + A²(2t)/2! + A³(3t²)/3! + …
which reduces to
d(exp(At))/dt = A + A²t + A³t²/2! + …

The matrix A can be factored as a premultiplier from each term so

d(exp(At))/dt = A[ I + At + A²t²/2! + …]

What is left in the brackets is none other than exp(At). Therefore

d(exp(At))/dt = A[exp(At)]

Now consider X(t) = exp(At)X₀. Differentiation by t shows that

dX/dt = A(exp(At)X₀) = AX(t)

Thus the function exp(At)X₀ satisfies the system of differential equations dX/dt = AX and the initial conditions.

Systems of Linear Inhomogeneous Equations

Consider a system of equations of the form

dX/dt = AX + C

where C is a vector of constants. If A has an inverse then the system can be converted into the form

dX/dt = A(X+D)

where D=A^-1C.

Let Y(t)=X(t)+D. Then the system becomes

dY/dt = AY

This system has the solution Y(t)=exp(At)Y(0) and thus

X(t) = exp(At)(X(0)+D) - D
or, equivalently
X(t) =exp(At)(X(0)+A^-1C) - A^-1C

Systems with Variable Coefficients

The systems previously considered all had constant coefficients. If any coefficient is a non-trivial function of time then the system

dX/dt = A(t)X

has the solution

X(t) = exp(∫₀^tA(s)ds)X(0)

If A(t) has an inverse for all t then the solution to the inhomogeneous system

dX/dt=A(t)X + C(t)
can be constructed as
X(t) = exp(∫₀^tA(s)ds)(X(0)+D(0)) - D(t)

where D(t)=A(t)^-1C(t).

Questions of Convergence

A square matrix M can be represented as

M = PΛP^T

Where P is an orthogonal matrix; i.e., P^-1=P^T; i.e., the inverse of P is equal to the transpose of P. The matrix Λ is a diagonal martrix. Thus

M^k = PΛ^kP^T

This means that

exp(M) = P*exp(Λ)P^T

The values of elements of Λ are called the eigenvalues of the matrix M and P is the matrix of its eigenvectors.

If
Λ=Diag(λ₁, λ₂, …, λ_n )
then
exp(Λ)=Diag(exp(λ₁), exp(λ₂), …, exp(λ_n) ).

One small complication is that the eigenvalues of a matrix may be nonreal. That is no major difficulty in that exp(x+iy) is well defined. It is exp(x)(cos(y)+isin(y)). A more serious complication occurs when an eigenvalue occurs with a multiplicity greater than unity. If an eigenvalue is repeated then the second occurence does not constitute another independent solution.

If an eigenvalue λ occurs with a multiplicity p then

{exp(λt), t*exp(λt), t²*exp(λt), …, t^p-1*exp(λt)}

are all solutions.

(To be continued.)

Higher Order Differential Equations

A differential equation of order k can be converted into a system of k first order equations. This just involves defining new variables such that

x_i = dx_i-1 /dt
for i=2 up to k

Systems of Nonlinear Differential Equations

Suppose a system of ordinary differential equations is representeed as

dX/dt = F(X)

One approach to a solution is by iteration. Start with an arbitrary X₀(t) then construct X₁(t) by integration of

dX₁/dt = F(X₀(t))

and likewise for X₂(t) and beyond as

dX_n+1/dt = F(X_n(t))

The question of convergence can be examined by looking at the differences y_n=X_n(t)−X_n-1(t). Then

dy_n/dt = F(X_n)−F(X_n-1)

The RHS of the above can be approximated by (∂F/∂X)·y_n(t). Thus

dy_n/dt = (∂F/∂X)·y_n(t)

The y's would go asymptotically to zero if the eigenvalues of the matrices (∂F/∂X) are all of a negative real part.

(To be continued.)