Download MAT282 * Differential Equations

Systems Of Linear Equations
Sec 10.1
Sec 10.2 – Basics Of Linear Systems
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The general form looks like this
y1 '  a11 y1  a12 y2  a13 y3  ...  a1n yn  f1 (t )
y2 '  a21 y1  a22 y2  a23 y3  ...  a2 n yn  f 2 (t )
...
yn '  an1 y1  an 2 y2  an 3 y3  ...  ann yn  f n (t )
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Which we can write
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And, the equation is homogeneous if all of the f’s = 0
Some Basic Operations (problems p 519)
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1. Verifying a solution
Some Basic Operations (problems p 519)
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2. System Of 3 equations
Some Basic Operations (problems p 519)
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3. With Initial Values
Some Basic Operations (problems p 519)
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4. A particular arrangement
Section 10.1 (cont.) Equations to Systems
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From a numerical standpoint it is sometimes useful
to rewrite an equation as a system
This procedure is pretty simple but requires some
care
The general idea is to, in a system of order n (so the
highest derivative is y (n)), write all of the lower
derivatives as new (unknown) functions and set it
up as a system.
Example
y (3)  6 y '' 3 y ' 2 y  3
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We simply give names to y, y’, y’’, then trace through
the relationships to reconstitute the original
equation
Let y0 = y
Let y1 = y’
Let y2 = y’’
Then y2 = y1’, and so on
The original equation becomes the system