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Unformatted text preview: Notes for Day : . . : Introduction to First Order Linear Systems A system of rstorder linear di erential equations has the form: y y y ( = = = a (t)y + b (t)y + c (t)y + g (t) a (t)y + b (t)y + c (t)y + g (t) a (t)y + b (t)y + c (t)y + g (t) is is an example with three functions; any number of functions are allowed in the system.) Solving a system of rstorder linear di erential equations requires nding a set of functions {y , y , y } which makes all of the equations in the system true simultaneously. Notation for linear systems
e above system of rstorder linear di erential equations can be written as an equation involving matrices: y y y a (t) a (t) a (t) b (t) b (t) b (t) c (t) c (t) c (t) y y y g (t) g (t) g (t) = + is textbook uses bold lowercase letters to indicate a column matrix and ITALIC CAPITAL letters to indicate a square matrix, and I'll follow that convention. Using the book's notation, we can rewrite the above system as: y = P(t)y + g(t). p. , Example Consider the scenario depicted below in which two gallon tanks are interconnected. Initially, Tank contains gallons of fresh water, and Tank contains gallons in which pounds of salt are dissolved. Let Q (t) and Q (t) denote the quantity of salt in each tank respectively at time t. Set up a system of equations which model Q and Q . Existence and Uniqueness
eorem . : Existence and Uniqueness Consider the initial value problem y = P(t)y + g(t) y(t ) = y , Let the n functions in P(t) and the n functions in g(t) all be continuous on the interval (a, b), and let t (a, b). en the intial value problem has a unique solution that exists on the entire interval (a, b). p. , Example Consider the initial value problem y = (sin t)y + t y + t  tt  y = (ln t + )y + e y + sec t y ( )= y ( )= Find the largest tinterval on which eorem . guarantees a unique solution to the problem exists. Rewriting an nth order linear equation as a rstorder system
It is possible to rewrite any nthorder linear equation as a rstorder system. is is a "reallife" important skill because rstorder systems lend themselves to accurate approximation techniques, and in real life, a very close approximation that is computationally "easy" may be just as good as an exact answer that is di cult to obtain. p. , Example Rewrite the thirdorder initial value problem y  t y + t y + y = e  as a system of rstorder equations.
t y( ) = , y ( ) = , y ( ) = p. , Exercise t y = t + as a system of rstorder equations. Rewrite the equation (cos t)y  t y + p. , Exercise Rewrite the system of secondorder equations: y z as a system of rstorder equations. = = y + y  z + z + t z + z  y + y  sin t ...
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 Spring '06
 EDeSturler
 Equations, Linear Systems

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