5.1-The Initial-Value Problems For Ordinary Differential Equations

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1 5.-The Iniil-Vlue Problems For Ordinry Differenil Equions Consider solving iniil-vlue problems for ordinry differenil equions: (*) y f, y, b, y. If we know he generl soluion y of he ordinry differenil equion (**) y f,y hen he soluion of he iniil-vlue problem is he funcion y h sisfies he differenil equion (**) nd he iniil-vlue y. For mny problems, he exc form of he generl soluion of (**) my no exis. In h cse, we wn o find n pproximion o he soluion of he iniil-vlue problem for n ordinry differenil equion given in (*). This chper, we will su pproximion mehods for solving iniil-vlue problems for ordinry differenil equions given in (*). In his secion, we will firs review bsic properies of iniil-vlues problems nd su wo pproximion mehods h genere sequence of funcions y k such h lim k y k y.. Lipschiz Condiion: Le R x,y ; x nd y re rel numbers. Definiion A funcion f,y is sid o sisfy Lipschiz condiion in he vrible y on se D in R if here exiss consn L such h f,y f,y L y y, whenever boh poins, y nd, y re in D. The consn L is clled Lipschiz consn for f. Exmple Le f,y y nd D,y, y. Does f sisfy Lipschiz condiion on D? If so, find is Lipschiz consn. Le, y nd, y be in D, i.e., is in, nd y is in,. Observe h f,y f,y y y y y y y y y y y y y. Becuse nd y y, we hve f,y f,y y y 4 y y. So, f sisfies Lipschiz condiion nd is Lipschiz consn is 4. Noe h he Lipschiz consn L is no unique, h is, for ny L 4, he inequliy f,y f,y L y y lso holds. So, in prcice, we wn o find L s smll s possible.. A Sufficien Condiion for Lipschiz Condiion: Theorem Suppose f,y is defined on convex se D in R. If consn L exiss wih, y L, for ll, y in D, hen f sisfies Lipschiz condiion on D in he vrible y wih Lipschiz consn L.

2 Exmple Le f,y y nd D,y, y. Does f sisfy Lipschiz condiion on D? If so, find is Lipschiz consn. The pril derivive of f wih respec o y is, y y. Becuse, y y y 4, f sisfies Lipschiz condiion wih Lipschiz consn 4. Exmple Le f,y sin y. Deermine if f sisfies Lipschiz condiion in D, y, y. Compue he pril derivive of f wih respec o y : cos y nd,, y cos y cos y 4. So, f sisfies Lipschiz condiion wih consn 4., y cos y cos y. Becuse 3. A Sufficien Condiion for he Uniqueness of Soluion of n Iniil-Vlue Problem: Theorem Suppose h D, y b, y nd h f, y is coninuous on D. If f sisfies Lipschiz condiion on D in he vrible y, hen he iniil-vlue problem y f, y, b, y hs unique soluion y for b. Exmple Deermine if he iniil - vlue problem y y e,, y hs unique soluion for. If so, find he soluion excly or numericlly. Le D, y, y. Check if f, y y e sisfies Lipschiz condiion in D. Becuse, f, y sisfies Lipschiz condiion in D nd herefore by Theorem he iniil-vlue problem hs unique soluion. Solve d y e : (i) Solve he homogeneous equion d y by seprion of vribles: y d, ln y ln C, eln y e ln C, y h e C C. (ii) Find priculr soluion of he homogeneous equion d Le y p A B C e. Then y e :

3 nd y p A B e A B C e A B e A B C e A B C e e A e A B A e B C B e Ce e A, B, C B, C The soluion is y p e nd he generl soluion is: y y h y p C e. (iii) Solve C by he iniil-vlue y : y C e, C e. The generl soluion: y e e. Exmple Deermine if he iniil - vlue problem y sin y,, y hs unique soluion for. If so, find he soluion excly or numericlly. Le D, y, y. Clerly, f, y sin y is coninuous on D. From n erlier exmple, we know f, y sin y sisfies Lipschiz condiion on D in he vrible y. By Theorem, we know he iniil-vlue problem hs unique soluion. Solve he iniil vlue problem: sin y, y. We cnno solve i excly. Here is d numericl soluion for his iniil-vlue problem: he grph of y where y sin y,, y 4. Picrd s Mehod: Picrd s mehod is mehod pproxime he soluion y of he iniil-vlue problem: y f, y, b, y by sequence of funcions y k where y k re funcions: Th is, y, Derivion: For k, y k f x, yk x dx, k,,... y f x, dx, y f x, y x dx,... 3

4 This implies f x,y x dx y x dx y x y y y. y k f x, yk x dx. Exmple Find pproximions y k of he soluion o he iniil-vlue problem: y, y, by Picrd s mehod. d For his problem, f,y y nd y. Then y y x dx x x x dx y k k! k. To find he rue soluion: () Homogeneous soluion: d () Priculr soluion: Le y p A B. Solve A nd B: x dx 6 3 y, y d, lny h C, y h e C Ce y p A, y p y implies A A B, his, A A B, A,B y p y y h y p Ce, y C, C, y. y blue - - y 4, red y 5, green... y 6, blck y x Exmple Find pproximions y nd y of he soluion o he iniil-vlue problem: 4

5 sin y, y, by Picrd s mehod. d For his problem, f,y sin y nd y. Then y xsin x dx dx. y xsin x dx sin. y 3 xsin x x x sin x dx? Exmple Find pproximions y 3 of he soluion o he iniil-vlue problem: y d, y, by Picrd s mehod. y, y x dx y x x x dx y 3 y 4 x 3 x 3 x3 4 x4 x5 x dx x blck - y,blue-y,red-y, green - y 3 5. Approxime he soluion by Tylor series: 5

6 Exmple Deermine he firs 5 erms in Tylor series expnsion x of he soluion o he iniil-vlue problem: y, y. d Le y y y y... n! y n n... nd y f,y y. y, y y y y, y y y y, y y Exmple Deermine he firs 5 erms in Tylor series expnsion of he soluion o he iniil-vlue problem: y y, y. Le y y y y... n! y n n... nd f,y y. y, y y yy 4yy, y 4 9 y 4 y 4yy 4 y 4 8y y 4 y y yy, y 4 4 y 3 Exercises:. Review he definiion of funcion f sisfying Lipschiz condiion. () Suppose we know h he funcion f,y sisfies he following inequliy f,y f,y e / y y, for whenever boh poins, y nd, y re in D,y 3, y. Show h f,y sisfies Lipschiz condiion nd give (s smll s possible) Lipschiz consn. (b) Suppose we know h he funcion f,y sisfies he following inequliy f,y f,y cos y y y, for whenever boh poins, y nd, y re in D,y 3, y. Show h f,y sisfies Lipschiz condiion nd give (s smll s possible) Lipschiz consn.. Review he sufficien condiion for funcion f sisfying Lipschiz condiion.. Show h ech of he following funcions sisfies Lipschiz condiion in y on he indiced se D. i. f,y y, D,y ; is in R, y ii. f,y y e y, D,y ;, 5 y 5 iii. f,y y y, D,y ;, y b. Le M. Show h he funcion f,y 4 4y sisfies Lipschiz condiion in y on he se D,y ; M, y is rel. Does f sisfy Lipschiz condiion in y on he se D,y ;, y is rel? 3. Review he sufficien condiion for n iniil-vlue problem y f,y, b, y o hve unique soluion. 6

7 Deermine if ech of he following iniil-vlue problems hs unique soluion. () y e y,, y (b) y y y /, 3, y, 4. Review Picrd s mehod. Use Picrd s Mehod o find he indiced y k.. Fin 3 for he iniil-vlue problem y y e, y. b. Fin for he iniil-vlue problem y y, y. 5. Review Tylor series for pproximing he soluion of n iniil-vlue problems. Find he firs five erms in Tylor series expnsion in for he soluion of he following iniil-vlue problem.. y y, y b. y e y, y c. y sin y, y 7