Section 4.1 - 4.2

Section 4.1 - 4.2 - Notes for Day : . . : Introduction to...

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Unformatted text preview: Notes for Day : . . : Introduction to First Order Linear Systems A system of rst-order 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 rst-order 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 rst-order 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 t-interval on which eorem . guarantees a unique solution to the problem exists. Rewriting an nth order linear equation as a rst-order system It is possible to rewrite any nth-order linear equation as a rst-order system. is is a "real-life" important skill because rst-order 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 third-order initial value problem y - t y + t y + y = e - as a system of rst-order equations. t y( ) = , y ( ) = , y ( ) = p. , Exercise t y = t + as a system of rst-order equations. Rewrite the equation (cos t)y - t y + p. , Exercise Rewrite the system of second-order equations: y z as a system of rst-order equations. = = y + y - z + z + t z + z - y + y - sin t ...
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