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# Note 15 for MATH 1314 with Professor Marks at UH

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This 8 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 12 views.

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Date Created: 02/06/15
Math 1314 Poppers Lesson 15 1 Write the general form of a function expressing exponential growth 2 Write the general form of a function expressing exponential decay 3 Find the rate of change of the exponential growth function 4 At the beginning of 2002 the population of a city was 13 million people At the beginning of 2006 the population was 148 million people Assume population growth is exponential a nd the growth constant k b write an exponential function expressing the population of the city as a function of time t in years where t 0 corresponds to the beginning of 2002 c use the function from part b to predict the population of the city at the beginning of 2010 d find a formula for the growth rate of the population e how fast will the population be increasing at the beginning of 2010 5 A radioactive substance has a half life of 324 days At the beginning of a study there are 783 grams of the substance present a find the decay constant k b write an exponential decay function expressing the amount of the substance present after t days c use the function from part b to predict the amount of the substance that will remain after 100 days d find a formula for the decay rate of the substance e how fast will the substance be decaying after 100 days Lecture 16 o 7 mmquot nVPavameters 1W He Wm M WM WW M mm m KVLuNnvmmzn Second Order Equations and Planar Systems Inhomogeneous equation 31 aty bty Ft 8 Homogeneous equation 3 aty mm 0 9 Inhomogeneous system x Atx Fat 10 where At 4305 a1t lIll Math 3331 Section 19470 Lecture 16 Section 46 Planar S stems Fundamental Set of Solutions and Fundamental Matrix Let y1ty2t be FSS for 9 3105 y2t X l yam y 2t l is PM of x Atx and 1 y t y2t X 1 w l 1t y1t l where W05 y1ty 2t y 1ty2t A matrix valued function Xt is a fundamental matrix for the system X Atx if and only if X AtX X0 Xt0 is invertible for some to I Jiwen He University of Houston Math 3331 Section 1940 Lecture 16 March 2 2009 3 7 W m Wequot We cm a Mom v0 mmmmg n m x z m x z mm x my x x 5 m be m uinn w x z A x 9 mm m wzm e v39gtltquotlt e vex4 e4 nsemng v av 09 vimzuhvmminn xpxXamw h Section 46 Plana S t m Variation of Parameters for Systems cont Set Xt1FCt Jdt a EJ v1t y2tdt V2t y1tdt recall that Wt y1ty t yty2t Math 3331 Section 19470 Lecture 16 March 2 2009 57 Particular Solution to System Particular solution of 10 mm Xo 58 y1tv1t y2tv2t y 1tv1t y 2tv2t First component is part sol of 8 312905 311tvlt y2tv2t 11105 fly2tFtWti dt where 11205 fy1tFtWt dt Math 3331 i Section 19470 Lecture 16 March 2 2009 6 7 Section 46 Planar S stems Example Ex Find particular solution of y I y tan t FSS Of y y 0 24105 Cost 2 I 2 9205 Sint W05 COS t I Sin t 1 v1t fsinttantdt sint lnsecttant 2120 fcosttantdt 2 cost gt ypt Costn secttant Math 3331 Section 19470 Lecture 1639 March 2 2009 7 7 Jiwen He University of Houston

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