Pólya conjecture

In mathematics, and in particular ordinary differential equations, a Green's matrix helps to determine a particular solution to a first-order inhomogeneous linear system of ODEs.

For instance, consider $x^{\prime }=A(t)x+g(t)$ where $x$ is a vector and $A(t)$ is an $n\times n$ matrix function of $t$, which is continuous for $t\in I,a\le t\le b$, where $I$ is some interval.

Now let $x^1(t),...,x^n(t)$ be $n$ linearly independent solutions to the homogeneous equation $x^{\prime }=A(t)x$ and arrange them in columns to form a fundamental matrix:

$X(t)=[x^1(t),...,x^n(t)].$

Now $X(t)$ is an $n\times n$ matrix solution of $X^{\prime }=AX$.

This fundamental matrix will provide the homogeneous solution, and if added to a particular solution will give the general solution to the inhomogenous equation.

Let $x=Xy$ be the general solution. Now,

$x^{\prime }=X^{\prime }y+X{y}^{\prime }$

$=AXy+X{y}^{\prime }$

$=Ax+X{y}^{\prime }.$

This implies $X{y}^{\prime }=g$ or $y=c+\underset{a}{\overset{t}{\int }}{X}^{-1}(s)g(s)ds$ where $c$ is an arbitrary constant vector.

Now the general solution is $x=X(t)c+X(t)\underset{a}{\overset{t}{\int }}{X}^{-1}(s)g(s)ds.$

The first term is the homogeneous solution and the second term is the particular solution.

Now define the Green's matrix $G_0(t,s)=\{\begin{array}{ll}0& t\le s\le b\\ X(t)X^{-1}(s)& a\le s<t.\end{array}$

The particular solution can now be written $x_p(t)=\underset{a}{\overset{b}{\int }}{G}_0(t,s)g(s)ds.$

External links

• An example of solving an inhomogeneous system of linear ODEs and finding a Green's matrix from www.exampleproblems.com.