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Section 2.1 Duhamel's Principle

Straight to the point....

Duhamel's Principle gives us a particular solution \(Y(t)\) to a nonhomogeneous ODE \(p(t) y'' + q(t) y' + r(t) y = g(t)\text{.}\)

Given a function \(S(t;s)\) (variable \(t\) and parameter \(s\)) that satisfies

\begin{align*} p(t) S''(t ; s) + q(t) S'(t; s) + r(t) S(t ; s) \amp = 0, \\ S(s;s) \amp = 0, \\ S'(s ; s) \amp = 1, \end{align*}

a particular solution to \(p(t) y'' + q(t) y' + r(t) y = g(t)\) is

\begin{align*} Y(t) \amp = \int_{t_0}^t S(t;s) \frac{g(s)}{p(s)} \ ds. \end{align*}

One thing to note here is that \(Y(t_0) = Y'(t_0) = 0\text{.}\)

Duhamel's Principle also gives us a particular solution to a nonhomogeneous ODE of the form \(p(t) y'' + q(t) y' + r(t) y = g(t)\text{.}\) However, it's often a lot easier to obtain this particular solution through Duhamel's principle than it is to obtain it using Variation of Parameters.

Formally, say that we have a function \(S(t ; s)\) (variable \(t\) and parameter \(s\)) that solves the following system:

\begin{align*} p(t) S''(t ; s) + q(t) S'(t; s) + r(t) S(t ; s) \amp = 0, \\ S(s;s) \amp = 0, \\ S'(s ; s) \amp = 1. \end{align*}

The notation \(S(t ; s)\) means that we treat \(S\) not as a two-variable function, but as a one-variable function in \(t\) with a parameter \(s\text{.}\)

Following this, we use \(S'(t;s)\) as shorthand for

\begin{align*} S'(t;s) \amp = \frac{d}{dt} S(t;s). \end{align*}

For example, we might set \(S(t ; s) = t^s\text{,}\) for some constant \(s\text{.}\) Then, \(S'(t;s) = st^{s-1}\) and \(S''(t;s) = s(s-1)t^{s-2}\text{.}\)

Then, the particular solution of \(p(t)y'' + q(t)y' + r(t)y = g(t)\) that Duhamel's principle gives us is

\begin{align*} Y(t) \amp = \int_{t_0}^t S(t ; s) \frac{g(s)}{p(s)} \ ds. \end{align*}

This is all a lot to absorb. The first thing to notice is that \(S(t;s)\) is a solution of the homogeneous system \(p(t) y'' + q(t)y' + r(t)y = 0\text{.}\)

Let's take \(\{y_1,y_2\}\) to be a set of fundamental solutions of the homogeneous ODE. Then, since \(S(t;s)\) is a solution of the homogeneous ODE,

\begin{align*} S(t;s) \amp = c_1y_1(t) + c_2y_2(t). \end{align*}

This was the info extracted from \(p(t) S''(t ; s) + q(t) S'(t; s) + r(t) S(t ; s) = 0\text{.}\)

Looking now at the other conditions, \(S(s;s) = 0\) and \(S'(s;s) = 1\text{,}\) notice that these two give us the following two equations.

\begin{align*} c_1y_1(s) + c_2y_2(s) \amp = 0, \\ c_1y_1'(s) + c_2y_2'(s) \amp = 1. \end{align*}

Here, \(y_1(s)\) and \(y_2(s)\) are both knowns and the only unknowns in the above are \(c_1,c_2\text{.}\) This is just begging us to write the system above with a matrix. Let's do that:

\begin{align*} \bmat{y_1(s)}{y_2(s)}{y_1'(s)}{y_2'(s)} \bvec{c_1}{c_2} \amp = \bvec01. \end{align*}

Solving this isn't too bad:

\begin{align*} \bvec{c_1}{c_2} \amp = \bmat{y_1(s)}{y_2(s)}{y_1'(s)}{y_2'(s)}^{-1} \bvec01\\ \amp = \frac1{y_1(s)y_2'(s) - y_1'(s)y_2(s)} \bmat{y_2'(s)}{-y_2(s)}{-y_1'(s)}{y_1(s)} \bvec01\\ \amp = \frac1{y_1(s)y_2'(s) - y_1'(s)y_2(s)} \bvec{-y_2(s)}{y_1(s)}. \end{align*}

A lot of things look pretty familiar here. The fraction above has a denominator equal to \(y_1(s)y_2'(s) - y_1'(s)y_2(s)\text{,}\) which is exactly the Wronskian \(W[y_1,y_2](s)\text{.}\)

Another thing to note is that, from the solution of \(c_1,c_2\) above, it's clear that both of these depend on \(s\text{.}\) So, let's write \(c_1 = c_1(s)\) and \(c_2 = c_2(s)\text{.}\) This way, we now have

\begin{align*} S(t;s) \amp = c_1(s)y_1(t) + c_2(s) y_2(t), \end{align*}

where

\begin{align*} c_1(s) = -\frac{y_2(s)}{W[y_1,y_2](s)} \amp \quad \text{and} \quad c_2(s) = \frac{y_1(s)}{W[y_1,y_2](s)}. \end{align*}

The familiarity of the above should become much more pronounced when we plug this into \(Y(t) = \int_{t_0}^t S(t;s) g(s)/p(s) \ ds\text{.}\)

\begin{align*} Y(t) \amp = \int_{t_0}^t \Big[c_1(s)y_1(t) + c_2(s) y_2(t)\Big] \frac{g(s)}{p(s)} \ ds \\ \amp = \int_{t_0}^t \frac{y_1(s)y_2(t) - y_1(t) y_2(s)}{W[y_1,y_2](s)} \cdot \frac{g(s)}{p(s)} \ ds. \end{align*}

Hey! This is Variation of Parameters exactly. So, Duhamel's principle is like a “re-skin” of Variation of Parameters. However, as we'll soon see with some examples, it can be a lot easier to work with than variation of parameters.