## Systems of Differential Equations :::{.definition title="Wronksian"} For a collection of $n$ functions $f_i: \RR^n \to \RR$, define the $n\times 1$ column vector \[ W(f_i)(\vector p) \da \begin{bmatrix} f_i(\vector p) \\ f_i'(\vector p) \\ f_i''(\vector p) \\ \vdots \\ f^{(n-1)}(\vector p) \end{bmatrix} .\] The **Wronskian** of this collection is defined as \[ W(f_1, \cdots, f_n)(\vector p) \da \det \begin{bmatrix} \vertbar & \vertbar & & \vertbar \\ W(f_1)(\vector p) & W(f_2)(\vector p) & \cdots & W(f_n)(\vector p)\\ \vertbar & \vertbar & & \vertbar \\ \end{bmatrix} .\] ::: :::{.proposition title="Wronskian detects linear dependence of functions"} A set of functions $\theset{f_i}$ is linearly independent on $I \iff \exists x_0 \in I: W(x_0) \neq 0$. ::: :::{.warning} $W \equiv 0$ on $I$ does *not* imply that $\theset{f_i}$ is linearly dependent! Counterexample: $\theset{x, x+x^2, 2x-x^2}$ where $W \equiv 0$ but $x+x^2 = 3(x) + (2x-x^2)$ is a linear combination of the other two functions. ::: \todo[inline]{Sufficient condition: each $f_i$ is the solution to a linear homogeneous ODE $L(y) = 0$.} ### Linear Equations of Order $n$ The standard form of such equations is \begin{align*} y^{(n)} + a_1y^{(n-1)} + a_2y^{(n-2)} + \cdots +a_ny'' + a_{n-1}y' + y = F(x). \end{align*} All solutions will be the sum of the solution to the associated homogeneous equation and a single particular solution. In the homogeneous case, examine the discriminant of the characteristic polynomial. Three cases arise: \begin{enumerate} \item $D>0 \Rightarrow$ 2 Real solutions, $c_1e^{r_1x} + c_2e^{r_2x}$ \item $D=0 \Rightarrow$ 1 Real, 1 Complex, $(c_1 +c_2x)e^{r_1x}$ \item $D<0 \Rightarrow$ 2 Complex, $e^{ax}(c_1\cos bx + c_2\sin bx)$ \end{enumerate} That is, every real root contributes a term of $ce^{rx}$, while a multiplicity of $m$ multiplies the solution by a polynomial in $x$ of degree $m-1$. Every pair of complex roots contributes a term $ce^r(a\cos \omega x + b\sin \omega x)$, where $r$ is the real part of the roots and $\omega$ is the complex part. In the nonhomogeneous case, assume a solution in the most general form of $F(x)$, and substitute it into the equation to solve for constant terms. For example, \begin{enumerate} \item $F(x) = P(x^n) \Rightarrow y_p = a+bx+cx^2+\cdots+(n+1)x^n$ \item $F(x) = e^x \Rightarrow y_p = Ae^x$ \item $F(x) = A\cos (\omega x) \Rightarrow y_p = a\cos(\omega x) + b\sin(\omega x)$ \end{enumerate} \subsection{Annihilators} Use to reduce a nonhomogeneous equation to a homogeneous one as a polynomial in the operator $D$. \begin{enumerate} \item $(D-a) \Rightarrow e^{ax}$ \item $(D-a)^{k+1} \Rightarrow x^k e^{ax}, x^{k-1}e^{ax}, \cdots, e^{ax}$ \item $D^{k+1} \Rightarrow x^k, x^{k-1}, \cdots,C$ \item $D^2-2aD+a^2+b^2 \Rightarrow e^{ax}\cos(bx), e^{ax}\sin(bx)$ \item $(D^2-2aD+a^2+b^2)^{k+1} \Rightarrow x^k e^{ax}\cos(bx), x^{k-1} e^{ax}\cos(bx), x^k e^{ax}\sin(bx), x^{k-1}e^{ax}\sin(bx),\cdots$ \end{enumerate} \subsection{Complex Solutions} $F(x)$ of the form $e^{ax}sin(kx)$ can be rewritten as $e^{(a+ki)x}$