examples152-ch9

# examples152-ch9 - MATH 152 Fall 2006-07 Applied Linear...

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Unformatted text preview: MATH 152 Fall 2006-07 Applied Linear Algebra & Differential Equations Worked Examples Dr. Tony Yee Department of Mathematics The Hong Kong University of Science and Technology September 1, 2006 ii Contents Table of Contents iii 1 Introduction 1 2 First-Order Differential Equations 5 3 Second-Order Linear Equations 39 4 Laplace Transform 83 5 Matrix 127 6 Systems of Linear Equations 135 7 Euclidean Vector 155 8 Eigenvalue and Eigenvector 177 9 Systems of Differential Equations 209 10 Orthogonality 231 iii Chapter 9 Systems of Differential Equations ¥ Example 9.1 (Converting higher-order equations to a system of first-order equations) Rewrite the equation as a system of first-order equations and find its corresponding initial value problem. ( u 00 + 0 . 125 u + u = 0 , u (0) = 1 , u (0) = 2 . Solution Let x 1 = u and x 2 = u , we have one first-order equation x 1 = x 2 . By the assumption, we also have u 00 = x 2 . From the original equation, we have x 2 +0 . 125 x 2 + x 1 = 0, or x 2 =- x 1- . 125 x 2 . Therefore the system becomes ( x 1 = x 2 , x 2 =- x 1- . 125 x 2 . The initial conditions are ( x 1 (0) = u (0) = 1 , x 2 (0) = u (0) = 2 . 2 ¥ Example 9.2 (Converting a system of first-order equations to a higher-order equation) Rewrite the system as a second-order equation and find its corresponding initial value problem. 8 > < > : x 1 = x 2- 2 x 1 , x 2 = x 1- 2 x 2 , x 1 (0) =- 2 , x 2 (0) = 1 . Solution From the first equation, we have x 2 = x 1 + 2 x 1 . So, x 2 = x 00 1 + 2 x 1 . Putting it into the second, we have x 00 1 + 2 x 1 = x 1- 2( x 1 + 2 x 1 ). Therefore the second-order equation is x 00 1 + 4 x 1 + 3 x 1 = 0 . Now consider the initial conditions. From the first equation, we have x 1 (0) = x 2 (0)- 2 x 1 (0) = 1- 2 · (- 2) = 5 . Therefore the initial conditions are x 1 (0) =- 2 , x 1 (0) = 5 . 2 209 9. Systems of Differential Equations ¥ Example 9.3 (Modeling with differential system) Consider the two interconnected tanks shown in the figure. Tank 1 initially contains 30 gal of water and 25 oz of salt, while Tank 2 initially contains 20 gal of water and 15 oz of salt. Water containing 1 oz/gal of salt flows into Tank 1 at a rate of 1 . 5 gal/min. The mixture flows from Tank 1 to Tank 2 at a rate of 3 gal/min. Water containing 3 oz/gal of salt also flows into Tank 2 at a rate of 1 gal/min (from the outside). The mixture drains from Tank 2 at a rate of 4 gal/min, of which some flows back into Tank 1 at a rate of 1 . 5 gal/min, while the remainder leaves the system. Tank 1 Tank 2 Q 2 ( t ) oz salt 20 gal water Q 1 ( t ) oz salt 30 gal water 1 . 5 gal/min 1 oz/gal 1 gal/min 3 oz/gal 3 gal/min 1 . 5 gal/min 2 . 5 gal/min (a) Let Q 1 ( t ) and Q 2 ( t ) , respectively, be the amount of salt in each tank at time t . Write down differential equations and initial conditions that model the flow process. Observe that the system of differential equations is nonhomogeneous....
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## This note was uploaded on 09/30/2010 for the course MATH MATH152 taught by Professor Kcc during the Spring '10 term at HKUST.

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examples152-ch9 - MATH 152 Fall 2006-07 Applied Linear...

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