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3f-fall2010-chapter71

3f-fall2010-chapter71 - Linear Systems of First-Order...

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Linear Systems of First-Order Equations Laney College, Fall 2010 Fred Bourgoin We will only consider systems of two equations, but the methods presented here can be generalized to larger systems. There are more elegant methods, but many rely on more sophisticated linear algebra than is required for this course. Laplace transforms can also be used to solve linear systems of equations; an example is the extra-credit problem from our third exam. 1. Introduction A linear system of two first-order equations consists of two equations: braceleftBigg x prime 1 = p 11 ( t ) x 1 + p 12 ( t ) x 2 + g 1 ( t ) x prime 2 = p 21 ( t ) x 1 + p 22 ( t ) x 2 + g 2 ( t ) . If g 1 ( t ) = g 2 ( t ) = 0 for all t , the system is said to be homogeneous. Example: two-mass, three-spring systems. In the diagram, F 1 ( t ) and F 2 ( t ) are forcing functions. If we consider the masses separately, we get the following equations: braceleftBigg m 1 x primeprime 1 = - k 1 x 1 + k 2 ( x 2 - x 1 ) + F 1 ( t ) m 2 x primeprime 2 = - k 3 x 2 - k 2 ( x 2 - x 1 ) + F 2 ( t ) This is of course a system of two second-order linear equations. Example: higher-orderequations. Suppose we have the second-order linear equa- tion y primeprime + p 1 ( t ) y prime + p 2 ( t ) y = g ( t ) . 1
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If we let x 1 = y and x 2 = y prime , then the equation can be written as the following linear system of first-order equations: braceleftBigg x prime 1 = x 2 x prime 2 = - p 2 ( t ) x 1 - p 1 ( t ) x 2 + g ( t ) . Example: parallel LRCcircuits. Consider the circuit depicted in the diagram. If V is the voltage drop across the capacitor and I is the current through the inductor, then braceleftbigg dI / dt = 1 L V dV / dt = - 1 C I - 1 RC V , where L is the inductance, C is the capacitance, and R is the resistance. Example: interconnected tanks. Consider the two tanks in the diagram. Considering the tanks separately, we get the equations braceleftBigg dQ 1 / dt = - 3 Q 1 30 + 1 . 5 Q 2 20 + 1 . 5 dQ 2 / dt = 3 Q 1 30 - 4 Q 2 20 + 3 . 2. Just Enough Linear Algebra Recall that we are only interested in linear systems of two equations here. We will therefore restrict our attention to 2 × 2 matrices. It is somewhat of a crime since linear algebra is an extremely useful area of mathematics, but we are pressed for time. I highly recommend you take at least an introductory course in linear algebra, if you have not already done so. An m × n matrix A is a rectangular array of numbers with m rows and n columns. The number in the i th row and j th column is called the i,j - entry of 2
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A and is denoted by a ij or a i,j . Example: The 2,3-entry of the 2 × 4 matrix A = parenleftbigg 0 1 2 4 5 6 7 8 parenrightbigg is a 2 , 3 = 7. Two matrices of the same size are equal if their corresponding entries are equal; i.e., A = B iff a ij = b ij for all i and j . Matrices of different sizes cannot be equal. A matrix with all zero entries is called a zero matrix. The n × n (square) identity matrix I n is defined by a ii = 1 and a ij = 0 when i negationslash = j . The 2 × 2 identity matrix is I = parenleftbigg 1 0 0 1 parenrightbigg .
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