Mah 2250-004 Week 4 April 6-20 secions 7.-7.3 firs order sysems of linear differenial equaions; 7.4 mass-spring sysems. Mon Apr 6 7.-7.2 Sysems of differenial equaions (7.), and he vecor Calculus we need o sudy hem (7.2). Every differenial equaion or sysem of differenial equaions can be convered ino a firs order sysem of differenial equaions (7.). Announcemens Warm-up Exercise
Summary and coninuaion of Friday overview/inroducion o Chaper 7, which is abou sysems of differenial equaions. We began wih a specific wo-ank inpu-oupu model, wih he goal of racking he vecor of solue amouns in each ank. The iniial value problem for his ank sysem was of he form x = A x x 0 = x 0
The more general firs order sysem of differenial equaions, and associaed iniial value problem is, x = F, x x 0 = x 0 Exisence and Uniqueness for soluions o IVP's The above IVP is a vecorized version of he scalar firs order DE IVP ha we considered in Chaper. In Chaper we undersood why (wih he righ condiions on he righ hand side), hese IVP's have unique soluions. There is an analogous exisenceuniqueness heorem for he vecorized version we sudy in Chaper 7, and i's believable for he same reasons he Chaper heorem seemed reasonable. We jus have o remember he geomeric meaning of he angen vecor x o a parameric curve in n (which is also called he velociy vecor in physics, when you sudy paricle moion) Algebra x lim 0 x n x 2 x n = lim 0 x x 2 2 x x n n = x n Geomeric inerpreaion in erms of displacemen vecors along a parameric curve So he exisence-uniquess heorem for firs order sysems of DE's is rue because if you know where you sar a ime 0, namely x 0 ; and if you know your angen vecor x a every laer ime -in erms of your locaion x and wha ime i is, as specified by he vecor funcion F, x ; hen here should only be one way he parameric curve x can develop. This is analogous o our reasoning in Chaper ha here should only be one way o follow a slope field, given he iniial poin one sars a.
For he wo-ank example, we used oupu from pplane, he siser program o dfield, o illusrae how soluions follow angen vecor fields, and racked he soluion o = 4 2 4 2 = 4 2 4 2 = 4 2 0 0 = 9 0 We noiced ha he limiing solue amouns appeared o be sense. = 3 6, which makes complee Noe This sysem of DE's is auonomous, since he formula x only depends on he value of x and no on he value of, so he angen field is no changing in ime; he pplane phase porrai is analgous o he phase diagram lines we drew for auonomous firs order differenial equaions in Chaper 2.
On Friday, we began o solve he wo-ank IVP example analyically, using eigenvalues and eigenvecors from he marix A in ha problem (!). Tha is ypical of wha we will do in secion 7.3, o solve he firs order sysem of DE's x = A x x 0 = x 0. Bu before we finish ha compuaion, i's a beer idea o review and exend some differeniaion rules you probably learned in mulivariable Calculus, when you sudied he calculus of parameric curves. This is relaed o maerial in secion 7.2 of he ex. ) If x = b is a consan vecor, hen x = 0 for all, and vise-verse. (Because all of he enries in he vecor b are consans, and heir derivaives are zero. And if he derivaives of all enries of a vecor are idenically zero, hen he enries are consans.) 2) Sum rule for differeniaion y d d x y = x y Boh sides simplify o y 2 x n y n 3) Consan muliple rule for differeniaion d d c x = c x Boh sides simplify o c. x n
4) Marix-valued funcions someimes show up and need o be differeniaed. This is done wih he limi definiion, and amouns o differeniaing each enry of he marix. For example, if A is a 2 2 marix, hen d d a a 2 a 2 a 22 = lim 0 a a 2 a 2 a 22 a a 2 a 2 a 22 = lim 0 a a a 2 a 2 a 2 a 2 a 22 a 22 = lim 0 a a a a 2 2 a a 2 2 a a 22 22 = a a 2 a 2 a 22. 5) The consan rule (), sum rule (2), and consan muliple rule (3) also hold for marix derivaives.
Universal produc rule Shorcu o ake he derivaives of f x (scalar funcion imes vecor funcion), f A (scalar funcion imes marix funcion), A x (marix funcion imes vecor funcion), x y (vecor funcion do produc wih vecor funcion), x y (cross produc of wo vecor funcions), A B (marix funcion imes marix funcion). As long as he "produc" operaion disribues over addiion, and scalars imes he produc equal he producs where he scalar is paired wih eiher one of he erms, here is a produc rule. Since he produc operaion is no assumed o be commuaive you need o be careful abou he order in which you wrie down he erms in he produc rule, hough. Theorem. Le A, B be differeniable scalar, marix or vecor-valued funcions of, and le produc operaion as above. Then be a d d A B = A B A B. The explanaion jus rewries he limi definiion explanaion for he scalar funcion produc rule ha you learned in Calculus, and assumes he produc disribues over sums and ha scalars can pass hrough he produc o eiher one of he erms, as is rue for all he examples above. I also uses he fac ha differeniable funcions are coninuous, ha you learned in Calculus. Here is one explanaion ha proves all of hose produc rules a once d d A B lim 0 A B A B = lim 0 A B A B A B A B = lim 0 A B A B lim 0 A B A B = lim 0 A B B lim 0 A A B = lim 0 A B B lim 0 A A B = A B A B.
I is always he case ha an iniial value problem for single differenial equaion, or for a sysem of differenial equaions is equivalen o an iniial value problem for a larger sysem of firs order differenial equaions, as in he previous example. (See examples and homework problems in secion 7.) This gives us a new perspecive on e.g. homogeneous differenial equaions from Chaper 5. For example, consider his overdamped problem from Chaper 5 x 7 x 6 x = 0 x 0 = x 0 = 4. Exercise 3a) Solve he IVP above, using Chaper 5 and characerisic polynomial. 3b) Show ha if x solves he IVP above, hen x, x 0 = 6 7 T solves he firs order sysem of DE's IVP 0 = 0 4 Use your work o wrie down he soluion o he IVP in 3b.. 3c) Show ha if, T solves he IVP in 3b hen he firs enry x solves he original second order DE IVP. So convering a second order DE o a firs order sysem is a reversible procedure.
3d) Compare he characerisic polynomial for he homogeneous DE in 3a, o he one for he marix in he firs order sysem in 3b. I's a mysery (for now)! x 7 x 6 x = 0 0 6 7 Picures of he phase porrai for he sysem in 3b, which is racking posiion and velociy of he soluion o 3a. From pplane, for he sysem From Wolfram alpha, for he underdamped second order DE in 3a.
A larger example of convering higher order DE's and sysems of DE's ino firs-order ones... Example Consider his configuraion of wo coupled masses and springs Exercise 4) Use Newon's second law o derive a sysem of wo second order differenial equaions for,, he displacemens of he respecive masses from he equilibrium configuraion. Wha iniial value problem do you expec yields unique soluions in his case? (See homework in secion 7.)