ADVANCED MATHEMATICS FOR ECONOMICS /2013 Sheet 3: Di erential equations

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1 ADVANCED MATHEMATICS FOR ECONOMICS - /3 Shee 3: Di erenial equaions Check ha x() =± p ln(c( + )), where C is a posiive consan, is soluion of he ODE x () = Solve he following di erenial equaions: (a) x = 3( 4)(x + ), con x() =. (b) x = 3 (c) dx =(x 3 + 4x 3, con x() =. y)d. (d) p x x =, con x() =. (e) x = p x +. (f) x = (x ). (g) x = x 3. 3 Solve he following di erenial equaions: (a) x +xx = (b) ( + x )d + ( + )xdx = (c) (sin x x sin )d + (cos + cos x)dx = (d) (x + e x )x = ( + xe x ) (e) (3 +4x)d +( +x)dx = 4 Solve he following di erenial equaions: (a) ( x)+(x + )x = (b) dx xd = e d (c) x +xx = (d) x +(x 3 )x = 5 Solve he following di erenial equaions: (a) x +x = e con x() =. (b) 3x x = con x() =. (c) (x x)=(+ )e. (d) x (e) e (x (f) x x = e. x )=. x + = e con x() =. 6 Solve he following di erenial equaions: (a) x = 3 x 3. (b) x = x3 (c) x = 3. p + x,wihy() = 5 3. x()( + ).

2 (d) ( +3x) d +( ) dx =. (e) (xy cos x) dx +(x ) dy =, wih y() =. (f) x +x = cos e,wihx() =. (g) x + x = e. 7 The equaion x + a()x = b()x n is a Bernoulli equaion. I is a linear equaion for n = or n =, bu i is no linear for n 6=,. Suppose ha his is he case. (a) Prove ha he change of variable y = x for y(). (b) Solve x +x = x 3, x() =. n ransforms he equaion ino a linear equaion 8 Draw he phase diagrams of each of he following equaions, find he equilibrium poins and sudy heir sabiliy. (a) x = g (x) =(x + )(x ) (x ). (b) x = g (x) =(x + )(x )(x ). 9 The Verhusl model of populaion dynamics in coninuous ime is analogous o he model in discree ime. I is supposed ha he rae growh is proporional o he produc of he populaion level imes he populaion remaining unil reach a sauraion level K. Thus, he populaion evolves as P () =kp()(k P ()), k,k >. Here, K is he carrying capaciy. Find P and draw some soluion rajecories. Suppose ha he populaion y of a cerain species of fish in a given area of he ocean is described by he logisic equaion y y = r y. K The resource is used for food. Suppose ha he rae a which fish are caugh, E(y), is proporional o he populaion y. Thus, we assume ha E(y) = Ey, wihe a posiive consan. Then he logisic equaion is replaced by y = r This equaion is known as Schaefer model. y y K (a) Show ha if E<r, hen here are wo equilibrium poins, y =, y > ; Ey. (b) Show ha y is unsable and y is asympoically sable. (c) A susainable yield Y of he fishery is a rae a which fish can be caugh indefiniely. I is defined as Ey. Find Y as a funcion of he e or E and graph he funcion (i is known as he yield e or curve). (d) Deermine E so as o maximize Y and hereby find he maximum susainable yield Y m. Five college sudens wih he flu reurn afer Chrismas Holidays o an isolaed campus of 5 sudens. If he rae a which his virus spreads is proporional o he number of infeced sudens y and o he number no infeced 5 y, solve he iniial value problem y = ky(5 y), y() = 5 o find he number of infeced sudens afer days if 5 sudens have he virus afer one day. How many sudens have he flu afer five days? Deermine he number of days required for half he campus o be infeced.

3 Answer he following quesions: (a) Form he homogeneous linear ODEs from heir characerisic equaion. r 3r + 5 = ; r(r + ) =. (b) Form he homogeneous linear ODEs from he roos of heir characerisic equaion. r =, r = 4; r =3 4i, r =3+4i. (c) Form he homogeneous linear ODEs from heir general soluion. C e + C e ; C e + C e ; e / (C sin + C cos ). 3 Find he soluion of he following equaions: (a) x +3x +3x +x = (b) x x x +x = (c) x 4x +5x x = (d) x x +5x = (e) x x + 5x = (f) y iv 3y +4y = (g) x +3x +3x +x =4 (h) x +9x = e (i) x 3x x = cos (j) x + x = sen (k) x 3x +x =( + )e 3 4 Find he soluion of he following equaions, considering he iniial condiions: (a) y 3y 4y = 3, such ha y() =, y () = (b) y + y y =, such ha y() =, y () = 5 An equaion of he form x + ax + bx =, where a and b are real consans, is called an Euler equaion. Show ha he subsiuion of he independen variable s = ln ransforms an Euler equaion ino an equaion wih consan coe ciens for he new dependen variable y(s) =x(e s ). As an applicaion, find he soluion of he equaion x 4x 6x = for >. 6 Suppose ha a risky asse X grows a an average exponenial rae of bu i is subjeced o random flucuaions of insananeous volailiy. Le V (x) be he value of a securiy ha collecs xd euros coninuously when he price of he sock is X = x. Supposing ha he risk free ineres rae in he economy is r<, i can be shown by arbirage reasonings ha he value of he sock V (x) saisfies he equaion he Euler equaion x V (x)+axv (x) rv (x) =x. Find he general soluion and pick up he economically sensible soluion among hese. 3

4 7 Le he demand and supply funcions for a single commodiy be given by D() = 4 4P () 4P ()+P (), S() = 6+8P (). We have assumed ha he demand depends no only on curren price, P, bu also in expecaions abou he firs and second variaion of prices, given by P and P, respecively. Assuming ha marke clears a every ime, i.e. D() = S(), deermine he pah of P. Deermine a linear relaion beween iniial condiions P () and P () such ha he soluion is bounded. 8 An enomologis is sudying wo neighboring populaions of red and black ans. She has esimaed ha he number of black ans is approximaely 6, and ha of red ans is 5,. The ans begin fighing and our enomologis observe ha a any ime, he number of ans killed of one populaion is proporional o he number of ans alive of he oher populaion. However, red ans are more aggressive han black ans in such a way ha heir e eciveness in he figh is quadruple ha of black ans. The observer receives a call o her mobile phone and mus leave he observaion, coming back o he camp. She knows ha hese wo species of ans figh unil one of hem is annihilaed. She conjecure ha, given ha he iniial populaion of ans is 4: in favor of blacks, bu he e eciveness is 4: for reds, boh populaions will pracically exinc a once. However, when she reurns nex day o he anhill, he siuaion is quie di eren. Could you help our hero o undersand wha happened by answering he following quesions? (a) Which is he survival species? (b) How many ans of he survival species remain alive? (c) Which should be he iniial proporion of boh populaions in order ha boh species become exinc a once? Hin: Denoing x() = black ans a ime, y() = red ans a ime (boh in housands), jusify why he ineracion beween ans can be given by x () = y () = 4ky(), kx(), wih k> a consan which is he figh e eciveness of black ans. This sysem can be convered ino a second order ODE for x() alone (or for y()). Then, solve and find he pahs of x() and y(), knowing ha x() = 6 and y() = 5. 9 Obain he general soluion of he following complee sysems of equaions. In Case (d) compue he paricular soluion considering he iniial condiions. (a) (b) (c) X X 8 < : A X A X x = x + y + z + e y = x y + z + e 3 z = x + y z +4 sen cos e e A A 4

5 (d) X = 3 e X + 3, X() = - Classify he equilibrium poin (, ) of he following sysems, in erms of he parameer. (a) X = X, ( 6= ). 6 3 (b) X = X. 3 The model of Obs of moneary policy in he presence of an inflaion adjusmen mechanism is as follows. The quoien M d /M s (money demand/money supply), is denoed by µ; p = P /P is he inflaion rae (P is he price level of he economy); q = Q /Q he consan (exogenous) rae of growh of GDP, Q, and m = M s/m s he moneary expansion rae. The evoluion of p follows he Walrasian adjusmen mechanism p = h( µ), <h< a parameer. Hence an excess in he moneary supply M s >M d, leads o a posiive incremen in he inflaion rae. To sipulae he ime evoluion of µ we consider he following assumpion: moneary demand is proporional o GDP in nominal erms, ha is, M d = ap Q, a > consan, hence Taking logarihms µ = a PQ M s. ln µ =lna +lnp +lnq ln M s, and aking he derivaive wih respec o ime we ge µ µ = P P + Q Q M s M s = p + q m. Hence, he sysem of ODEs in he model of Obs is p = h( µ), µ =(p + q m)µ. The exercise sudies he e ec of he moneary policy chosen by he cenral bank, given by m. (a) Suppose ha m = m is consan (exogenous and consan moneary expansion rae) and ha m>q. Show ha he sysem has a cener. (b) Suppose ha m = m p wih > (counercyclical convenional moneary policy) and m>q. Show by means of he phase porrai ha he qualiaive behavior of he sysem is similar o (a) above. (c) Suppose ha m = m p (Obs s Rule) wih >and m>q. Prove ha for some values of he sysem has a spiral aracor. (d) Wha do you hink abou he sabilizaion properies of he counercyclical rule and Obs s Rule? N. P. Obs (978) Sabilizaion policy wih an inflaion adjusmen mechanism. Quarerly Journal of Economics, May, pp