Content deleted Content added
→Description of method: Pulled equation numbering out of <math> tags and made them linkable |
m →General second-order equation: Delimited matrices with square brackets to visually distinguish from parentheses everywhere used for function calls |
||
Line 216:
We have the system of equations
:<math>\begin{
u_1(x) & u_2(x) \\
u_1'(x) & u_2'(x) \end{
\begin{
A'(x) \\
B'(x)\end{
\begin{bmatrix} 0 \\ f \end{bmatrix}.</math>
\begin{pmatrix}▼
Expanding,
:<math>\begin{
A'(x)u_1(x)+B'(x)u_2(x)\\ A'(x)u_1'(x)+B'(x)u_2'(x) \end{ = \begin{ So the above system determines precisely the conditions
Line 237 ⟶ 238:
We seek ''A''(''x'') and ''B''(''x'') from these conditions, so, given
:<math>\begin{
u_1(x) & u_2(x) \\
u_1'(x) & u_2'(x)
\end{ \begin{
A'(x) \\
B'(x)\end{
\begin{
0\\
f\end{
we can solve for (''A''′(''x''), ''B''′(''x''))<sup>T</sup>, so
:<math>\begin{
u_1(x) & u_2(x) \\
u_1'(x) & u_2'(x)
\end{ \begin{
u_2'(x) & -u_2(x) \\
-u_1'(x) & u_1(x) \end{
\begin{
where ''W'' denotes the [[Wronskian]] of ''u''<sub>1</sub> and ''u''<sub>2</sub>. (We know that ''W'' is nonzero, from the assumption that ''u''<sub>1</sub> and ''u''<sub>2</sub> are linearly independent.) So,
| |||