Question

Homework #6-Due Monday, March 11th Problem 1. Solve the IVP Problem 2. Solve the IVP Prolem 3. Solve the IVP Problem 4. Show that the set e-isto-rfundamestal wt f o Problem 5.Isa).ga)sina)) afulamestal set of solutiono with 0 x?Explain Problem 6. Consider the ODE and sappose that the oots for the characteristk equation+c0 ae the pair of Show that einco(Bt)is a fundamentall set of olutions for this ODE HINT: You will need the fact that given a quadratic ﹃watin ar' + ﹄ + c with pota η min the following hold Problem 7. Cosider the and suppose that the characteristic yuation or++cm0has only one oot and we call it r for the fundammental set of solutions unctlos et) that would give us the second solutlon Then we know that one of the solutions for this ODE is nt)e bt we ed wo Assume that the secotal siltion 9, ๒ of the kem MIt)-r(0'uddetrtir al Prove that ()e"velt)eltl" is in fat a fundaental set of Compare your findings with the Sornda we saw is da HINT: Define alt)ftfind the fint and weond derivatives of nformation in the eqaation and use the facts that this (1) We ate assuming h to be a solution (2) We are assaning r to be the ONLY root of the characteristic esation theeedoxe with on" alvira invoked you willainate everything and wilmd up with«To- From there you should be alle to conchde

Answer 1

"Given that y"" + y' - 2y = 0 y(0) = 1: y'(0) = 1 d^2 y/dx^2 + dy/dR - 2y = 0 write it in a Axillary Equation m^2 + m - 2 = 0 m^2 + 2m - m - 2 = 0 m(m + 2) - 1 (m + 2) = 0 (m + 2) (m - 1) = 0 m = 1, -2 Solution of the Equation is y = C_1 e^x + c_2 e^-2x y(0) = c_1 + c_2 = 1 y'(x) = c_1 e^x - 2 c_2 e^-2x c_2 = 0 solution is y = e^x \r\n Given that 16y"" - 8y' + 145 uy = 0 y(0) = -2 y'(0) = 1 write it in Auxiliary Equation 16m^2 - 8m + 145 = 0 roots are complex i.e, (m - 0.25 + 3i) (m - 0.25 - 3i) therefore m = 0.25 + 3i solution is y = e^3x [c_1 cos 0.25 x + c_2 sin 0.25 x] y(0) = c_1 = -2"