MATH 202 Final, Spring 2010

American University of Beirut “4 au MATH 202 Lota ‘ob \\ Differential Equations > “ Spring 2010 Final Exam Useful formulas are given on the backside of the paper PUT YOUR SECTION’S NUMBER ON YOUR BOOKLET PLEASE: Dr LYZZAIK: 1(F-ipm) 2(F-12pm) 3(F-10am) 4(F- 9am) Dr KOBEISS: 13(T-1lam) 14 (T- 12:30pm) 15 (T-2pm) 16 (T - 9:30am) Exercise 1 (10 points) Given that y = sinz is a solution of y) + 2y!” + 11y" + 2y! + 10y =0 find the general solution of the DE Exercise 2 (10 points) Solve the DE: x?y”+ay'—y=Inz_ on the interval (0, +00) 7 . Bxercise 3 (15 points) Find two power series solutions of the equation y" —2y' +y=0 ot the ordinary point z = 0 (give the radius of convergence of the series) Exercise 4 (20 points) Evaluate: . . (i) £{cos(2t)U(t — r)} (ii) £7? i 5} (iii) £ {ef sinr ar} Gy) {x + 5 - i} Exercise 5 (15 points) Use Frobenius method to find ONE series solution (corresponding to the larger root) of the equation eA x(x — 1)y” + 8y/ — 2y =0 about the regular singular point x = 0. Write the solution in concise form. Exercise 6 (10 points) Show that y = /zJ3 /2(z) is a solution of the differential equation wa xy" + (a? —2)y=0 Exercise 7 (10 points) Use the Laplace transform to solve the IVP hee { y" + 4y’ + 13y = 6(t — 2) + 6(t — 3m) y(0)=1, y/(0)=0 Exercise 8 (10 points) Find the general solution of the given system zs “= 82—y—z ba LY gtty z =ar-ytez dt good luck Bessel Equation: The solution of the Bessel equation \ 4 zy!" 4 zy! “ (x? = v*)y =0 ty isy =cJ,(z) + @Y,(z) forz>0 per +1)” xz\ 2n+v arena (a) The Bessel function J, = > aid tute n=0 Laplace Transform: copa [enna = LL F008} = Fs) — 8"17(0) — 2400) —... — FN (0) - Let f()} = F(s a) - £{ F(t a)U(t —a)} =e F(s) en as - L{U(t ~a)} = s - L{g(t)Ul(t — a)} =e *L{9(t +0)} - LAE F(t} = (-1)" (6) -fe9= f salt—ndr - LF x9} = LEP ILE} a“ (ne!) - L{5(t — to)} = e- sto