Question

Consider the ODE below. y' + 364 = sec(62) Find the general solution to the associated homogeneous equation. Use C1 and C2 as arbitrary constants. y(2) Use variation of parameters to find a particular solution to the nonhomogeneous equation. State the two functions vi and U2 produced by the system of equations. Let vi be the function containing a trig function and V, be the function that does not contain a trig function. You may omit absolute value signs and use 0 for any constants of integration. v1(2) = 022) What is the particular solution yp produced by variation of parameters? Yp(2) =

Answer 1

WI are VyLX) = su sa sin (62) secc62) are V, 1x) = 6 sin (6x) Cox 16x) obe VICH) = Vical = of tan (6x) che &(- ln I vier) -- In / cos 621 VI(2) = t en/ cos 6x/ cos (69) sec (6n) da and Ug (2) = 36 W2 dae 6 S A w L2€ 6 to fare 이가 T Veln) = cos (64) G 208 (63) Vacx) = 446 6 so Jo = 4, V1y2 Veq = cos (63) In cos 6re! tre sinone 36 Jp = 36 o cos (6x) lem/608(63) + x sin(62 and general solution is y = Ye typ y = gcos bretç.einot tot cor (69) An (605160) + fusInC) is solution this required

Solution Guiven differential equation es 36y= y "+ 36 = sec (62) Griven differential equation can be written as Co2+36)4= sec (62) the auxiliary equation is m2+36 50 m2 = 36 m=thi = C,COS 6-team sin 620 CF = Yc Here gi= cosbre Y = sin are are solution of homogeneous differential eqh o sinon colore y 42 حماMe 6 corox พ į Il W= 6 co8l6x + 6sin? cze = 6( sinkku teox?6 x) W=6 o sincere) and WI 0062 42 fox) Ya sec(60) 6 corox wy sin (62) sec (62) 208 6630) y fixe) -G sin 6x) secco a nza sec (67) Wa= Cos (6x) see (6x)