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# Note for MATH 3321 at UH

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Date Created: 02/06/15
CHAPTERl Introduction to Differential Equations 11 Basic Terminology Most of the phenomena studied in the sciences and engineering involve processes that change with time For example7 it is well known that the rate of decay of a radioactive material at time t is proportional to the amount of material present at time t In mathematical terms this says that d i ky k a negative constant 1 dt where y yt is the amount of material present at time t If an object7 suspended by a spring7 is oscillating up and down7 then Newton7s Second Law of Motion F ma combined with Hooke7s Law the restoring force of a spring is proportional to the displacement of the object results in the equation dw k270 k quot 2 g y 7 7 a pos1t1ve constant where y yt denotes the position of the object at time t The basic equation governing the diffusion of heat in a uniform rod of nite length L is given W Bu i 7 3 at 812 where u uzt is the temperature of the rod at time t at position I on the rod Each of these equations is an example of what is known as a differential equation DIFFERENTIAL EQUATION A di ferential equation is an equation that contains an unknown function together with one or more of its derivatives Here are some additional examples of differential equations Example 1 2 fryiy M yiegjfv d2 d b 12 7 21 2y 413 8211 8211 c w w 0 Laplace7s equation dy 7 dgy d2y 71 d 7 4E73e i TYPE As suggested by these examples7 a differential equation can be classi ed into one of two general categories determined by the type of unknown function appearing in the equation If the unknown function depends on a single independent variable then the equation is an ordinary di er ential equation if the unknown function depends on more than one independent variable then the equation is a partial di erential equationi According to this classi cation the differential equations 1 and 2 are ordinary differential equations and 3 is a partial differential equation In Exam ple 1 equations a b and d are ordinary differential equations and equation c is a partial differential equation Differential equations both ordinary and partial are also classi ed according to the highest ordered derivative of the unknown function ORDER The order of a differential equation is the order of the highest derivative of the unknown function appearing in the equation Equation 1 is a rst order equation and equations 2 and 3 are second order equations In Example 1 equation a is a rst order equation b and c are second order equations and equation d is a third order equation In general the higher the order the more complicated the equation In Chapter 2 we will consider some rst order equations and in Chapter 3 we will study certain kinds of second order equationsi Higher order equations and systems of equations will be considered in Chapter 6 The obvious question that we want to consider is that of solving a given differential equation SOLUTION A solution of a di erential equation is a function de ned on some interval I in the case of an ordinary differential equation or on some domain D in two or higher dimensional space in the case of a partial differential equation with the property that the equation reduces to an identity when the function is substituted into the equation Example 2 Given the secondorder ordinary differential equation 12 y 7 21 y 2y 413 Example 1 show that a 12 213 is a solution b 2x2 31 is not a solution SOLUTION a The rst step is to calculate the rst two derivatives of y y 12 213 y 21 612 y 2 121 Next we substitute y and its derivatives into the differential equation 122 121 7 2121 612 212 213 77 413 Simplifying the lefthand side we get 212 1213 7 412 712a3 212 413 i 413 and 413 7 413 The equation is satis ed y 12 213 is a solution b The rst two derivatives of 2 are 2 212 31 2 4x 3 ZN 4 Substituting into the differential equation we have 124 7 2141 3 2212 31 7 413i Simplifying the lefthand side we get 412781276z4z26z 0413 The function 2 212 31 is not a solution of the differential equation I Example 3 Show that uz y cos I sinh y sin I cosh y is a solution of Laplace s equation 8211 8211 w wwi SOLUTION The rst step is to calculate the indicated partial derivatives 7 7 sin I sinh y cos I cosh y g 7 cos I sinh y 7 sin I cosh y 7 cos I cosh y sin I sinh y gig cos I sinh y sin I cosh yr Substituting into the differential equation we nd that 7 cos I sinh y 7 sin I cosh y cos I sinh y sin I cosh y 0 and the equation is satis ed uz y cos I sinh ysin z cosh y is a solution of Laplace s equation I Exercises 1 1 ll Classify the following differential equations With respect to type iiel7 ordinary or partial and order a y2 zyy sin 1 b y em tan 1 8211 8211 8211 c w27azay er 70 d2 3 cl zy2zl e y 75zy y e 71 f BuBz MailBy d2y dy 2 ds 721 g gigy Jrry i le l For each differential equation determine Whether or not the given functions are solutions 2 y 4y 0 sin 31 cos 21 2sin 21 d3 d TZ61 yI1sinIe 7 zz2coszexl z z 4 zy y 0 y1r1n1I7 MI 12 5l11y zy 7y 1 12 e 12 17 121l d3 d2 d 6 E 7 ST 6 0 61621 02637 cl Cg constants7 262 3631 4 8211 8211 2 2 3 2 7i 770 u1zylnI y u2zyz 731yl 812 8y2 8 ylliy27z yze 172 21sinhz172l Bu 7 k2 8211 i 7 739 k at 812 k u1zt 6 2t cos I 11217 6 2t sin 27ml Find the set of all solutions of each of the following differential equations 10 y 21 lnz 11 y 32 12 y 61 cos 21 dyi 13 i73l dz y

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