Dtjytyk

If we have a nonhomogeneous system of linear differential equations:
$$frac{dvec{y}}{dx}=Avec{y}+vec{b}$$
where:
$$frac{dvec{y}}{dx}=begin{bmatrix}y_1'\y_2'\y_3'end{bmatrix},vec{y}=begin{bmatrix}y_1\y_2\y_3end{bmatrix},A=begin{bmatrix}a_{11}&a_{12}&a_{13}\a_{21}&a_{22}&a_{23}\a_{31}&a_{32}&a_{33}end{bmatrix}, vec{b}=begin{bmatrix}b_1(x)\b_2(x)\b_3(x)end{bmatrix}$$
the particular solution involving these three functions could be written:
$$y_{particular}=c_1(x)e^{lambda_1x}vec{u_1}+c_2(x)e^{lambda_2x}vec{u_2}+c_3(x)e^{lambda_3x}vec{u_3}$$
where $lambda_i, vec{u_i}$ are the eigenvalues and eigenvectors of the constant matrix $A$ (I chose it constant to make things simpler.)

When trying to evaluate these $c_i(x)$ functions, I was told to evaluate them by:
$$c_i(x)=intfrac{W_i}{W}dx$$
Where $W$ is the Wronskian determinant whose columns are the elements of the vectors $c_i(x)e^{lambda_3i}vec{u_i}$, and $W_i$ is the Wronskian determinant whose $i$th column was replaced by the contents of the $vec{b}$ vector.

My questions are: How does the integral show up? Why does the process look like Cramer's rule? How did we lead up to that result? Where did this 'Wronskian' determinant come from?

If I have missed something, please point it out and I'll add it to the post.

Thanks in advance.

Write the general solution of the homogeneous linear ODE
$$
tag{1}
frac{mathrm{d}vec{y}}{mathrm{d}x} = A(x) vec{y}
$$
in the matrix form as
$$
Phi(x; x_0) vec{c},
$$
where $vec{c} = mathrm{col}(c_1, dots, c_n)$ is a constant matrix (that is, independent of $x$), and $Phi$ is the state-transition matrix such that $Phi(x_0, x_0) = I$ (the identity matrix).

Now we are looking for a general solution of the nonhomogeneous linear ODE
$$
tag{2}
frac{mathrm{d}vec{y}}{mathrm{d}x} = A(x) vec{y} + vec{b}(x)
$$
in the form
$$
Phi(x, x_0) vec{c}(x).
$$
Incidentally, here is the source of the term "variation of constants". We look for conditions for the above function of $x$ to be a solution of $(2)$, that is, to have
$$
frac{mathrm{d}}{mathrm{d}x}(Phi(x, x_0) vec{c}(x)) = A(x) (Phi(x, x_0) vec{c}(x)) + vec{b}(x).
$$
But
$$
frac{mathrm{d}}{mathrm{d}x}(Phi(x, x_0) vec{c}(x)) = frac{mathrm{d}}{mathrm{d}x} Phi(x, x_0) cdot vec{c}(x) + Phi(x, x_0) cdot vec{c}'(x)
$$
and
$$
frac{mathrm{d}}{mathrm{d}x} Phi(x, x_0) = A(x) Phi(x, x_0),
$$
so after cancellation we obtain
$$
Phi(x, x_0) cdot vec{c}'(x) = vec{b}(x),
$$
which is, for each fixed $x$, a Cramer (because $det{Phi(x, x_0)} = W(x, x_0) ne 0$) system of $n$ linear equations with $n$ unknowns, $c'_1(x), dots, c'_n(x)$. The unknowns are given by the Cramer rule. We have thus obtained the derivative of the matrix function $vec{c}(x)$, so we have to integrate that derivative. And that is the source of the integral.

The "Wrońskian" determinant was named after a nineteenth-century Polish philosopher and mathematician, Józef Maria Hoene-Wroński.

EDIT: I changed slightly the terminology and provided a reference to a Wikipedia entry.