Question

Do just the marked ones (21,22,25)

PART III. Given these pairs of differential operators, what operator is the com- sition of the operators? Multiply the two characteristic polynomials and construct the composite operator by inspection. L20) y" y 23. L1()" - 5y 24. Li (y) = y" + y; L2() +2y L2(y) = y" + y PART IV. Nonhomogeneous differential equations with constant coefficients. For the following differential equations, factor the characteristic polynomials of the op- erator and of an annihilator of the right hand side. Multiply these two polynomials in factored form. By inspection write down a kernel of the composite operator that corresponds to the product polynomial and delete from this kernel the kernel of the operator of the differential equation. Finally, state the form to assume for a candidate particular solution by undetermined coefficients. 25. y" 4y 12y 15e- 4 cosx 13 sin x.

Answer 1

L_1 (y) = y' - 4y, L_2 (y) = y'' + y Two characteristic polynomials are (m - 4) and (m^2 + 1) = m^3 + m - 4m^2 - 4 After multiplying (m - 4) (m^2 + 1) = m^3 + m - 4m^2 - 4 Hence the composition of the operators is L(y) = y''' - 4y'' + y' - 4y \r\n L_1 (y) = y'' + y' - 4y, L_2 (y) = y'' + y' two characteristics polynomials are = m^4 + m^3 + m^3 + m^2 - 4m^2 + 4m = m^4 + 2m^3 - 3m^2 - 4m Hence the operator is L(y) = yiv + 2y''' - 3y'' - 4y' \r\n y'' - 4y' - 12y = 15 e^x - 4 cos x - 13 sin x characteristic polynomial is m^2 - 4m - 12 = 0 m^2 - 6m + 12m - 12 = 0 m(m - 6) + 2(m - 6) = 0 (m - 6) (m + 2) = 0 Kernel of the operator is m = 6, -2 Therefore y_c (x) = c_1 e^6x + c_2 e^-2x Hence particular solution is of the form y_p (x) = Ae^x + B cos x + C sin x