Question

Consider the following differential equation to be solved by variation of parameters.

y'' + y = csc(x)

Find the complementary function of the differential equation.

y_c(x) =

Find the general solution of the differential equation.

y(x) =

Answer 1

- Som Here we have differential equation y"+y = cosec.com) -0 We find complementary solution as follow Auxillany equation is me + 1 0 -> ma ti - Complementary solution is Ye (m) = . Casow) + q SINC)) — * Now we find general solution as follow We fined particular solution of egn © by variation af parameter Take Y, (n) = cos(u) (n) = sin(n) , g(x) = cosec (4) W(y14) = 1 so WCY, Yp (M) = -4,5 kg(n) du + y l rigon du J W14,1%) -- cosm J sincm cosec Coll dom + sinem I cos(a) . Case {wdex - = - cos (x) / sinema che + sinku s costume de Anw) = - Cos (n) + sin(n) fog (sin (D) Thus is the general solution y (m) - Y(u) + Yp(w) Y(") = Gos(x) + q sin(a) -n. Cos (n) + sin Above is our solution