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# Differential Equations MATH 308

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This 4 page Class Notes was uploaded by Maximilian Turcotte on Monday October 5, 2015. The Class Notes belongs to MATH 308 at Colgate University taught by Staff in Fall. Since its upload, it has received 45 views. For similar materials see /class/219591/math-308-colgate-university in Mathematics (M) at Colgate University.

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Date Created: 10/05/15

Math 308 This Maple session contains examples that show how to solve certain second order constant coefficient differential equations in Maple Also at the end the quotsubsquot command is introduced First we solve the homogeneous equation yquot 2y 5y 0 We39ll call the equation quoteqlquot gt eql diffyttt 2diffytt 5yt 0 13 2 J2 a J 5 0 eq r BIZYU aIW yt We use the quotdsolvequot command to solve the differential equation In its basic form this command takes two arguments The first is the differential equation and the second is the function to be found We39ll use the quotrhsquot command to save the actual solution in the variable quotsollquot gt 3011 rhs dsolve eqlyt Solving Second Order Differential Equations 30117C1 e t s1n2 t 7C2 e t cos2 t The two expressions 7C1 and 7C2 are Maple39s quotarbitrary constantsquot That was easy enough How do we specify initial conditions Consider the initial value problem yquot 2y 5y 0 yO3 y 05 We use the quotdsolvequot command again but we now make a list of the equation and the initial conditions The first initial condition y0 3 is written in Maple just as it is here However to enter the initial value of y we can not simply write y3905 The single quote 39 has a special meaning in Maple and it is not a derivative Instead we use the quotDquot operator The operator quotDquot is another way to specify the derivative of a function The derivative y39t can be expressed in Maple as Dyt quotDyquot means quotthe derivative of yquot so quotDytquot means quotthe derivative of y evaluated at tquot To specify the initial condition y 05 we use quotDyO5quot We use this in the following command gt sollivp rhs dsolve eqly0 3D y 0 5yt sollivp eH sin2 t 3 eH cos2 t Note that the first argument to quotdsolvequot is a list of three elements eqly03Dy05 This is the initial value problem to be solved As before the second argument is just yt the function to be found Plot the solution gt plot sollivp l 6 labels quottquot quotyquot 17 Now let39s try a nonhomogeneous example yquot 2y 5y 3sin2t gt eq2 diffyttt 2diffytt 5yt 3Sin2t 23 2 J2 a J 5 339 2 eq r aIZW aIW yt Sln t Use quotdsolvequot to nd the general solution 7 gt 3012 dsolveeq2yt it it 3 12 3012 yt e s1n2 t 7C2 e cos2 t 7C1 Es1n2 t Ecos2 t E Now solve an initial value problem We39ll use the same initial conditions as before gt solZiV39p rhs dsolveeq2y0 3Dy 05yt i it 14 63 it sol2z39v e s1n2t e l7 1 2t 3 39 2t 12 21 cos s1n cos 7 17 17 7 gt p10tsolZivptO 15 3 25 We see the initial transient dies out and the steadystate behavior is a sinusoidal oscillation You can i look back at the solution to see the terms that make up the particular solution Here s an equation with a more complicated function on the right i yquot y 2y t 2cos4t gt eq3 diff yt t t diff yt t 2yt t32cos 4t 37 3 2 a J 2 t2 4 eq r BIZYU aIW yt COS t 7 gt 3013 rhsdsolveeq3yt 1 7 1 3013e 12tsinE7IJ7C2e 12 cosE 7IJ7CJ 411281 10906 11236 f sin4 0 cos4 t18974t 39326 2 16123 7 595508 595508 Look carefully at the above solution you should be able to determine which terms are part of the homogeneous solution and which are part of the particular solution Let39s solve the problem with initial conditions y00 and y 00 and then plot the result 7 gt sol3ivp rhs dsolve eq3y0 OD y 0 0yt 33177 7 1 16123 7 1 3013 in 1 2 sin517t17 1 2t cos 17t e e 4168556 595508 1 41128 t 10906 11236 f sin4 t cos4 t 18974 t 39326 2 16123 1 595508 595508 7 gt plotsol3ivpt0 12 2 2 4 V Maple can solve differential equations that contain parameters 7 gt eq4 diffyt tt 4diffyt t kyt 0 43 2 J 4 J k 0 7 eq r BIZYU aIW W gt 3014 rhsdsolveeq4yt 7247k4t 72747k4t C e 7C2e sol4 Note however that Maple has only given the solution that we expect when k lt 4 Ifk4 then Maple s solution is not complete it is missing the term t exp2t and if k lt 4 Maple39s answer will become complexvalued and we want realvalued solutions If Maple knows the numerical value of k it will give the correct form of the answer To show this I ll first introduce a useful function called quotsubsquot i The quotsubsquot command short for quotsubstitutequot allows you to replace variables in an expression with other variables or values For example gt p x 34x 2 xy y 2 1 i px34xz xyy2l To nd the value of this expression when x3 we can use the following command gt subs x3 p i 64 3 y y2 The last argument of quotsubsquot is the expression in which to make the substitutions The other arguments have the form variable lalue More than one subsitution can be given in the command gt subs x3 y5p i 74 Now we39ll use the quotsubsquot command to create copies of eq4 with several different numerical values of l k gt eq4a subs kl eq4 4 82 42 J 0 t t I W a atz aty y gt dsolveeq4ayt 070 eltlt72 gtrgtic2eltiltz gtrgt gt eq4b subsk4eq4 gt eq4 7 tzyt EtyU 570 dsolveeq4byt 72 72 yt7C1e 07C2elt H Notice that Maple gave the correct answer here gt eq4c subsk8eq4 4 32 42 J 8 0 e c t t t q aim aty y gt dsolveeq4cyt 2 t 2 t yt7C1 e s1n2 t7C2e cos2 t And here it gave the correct realvalued solution

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