Question

Find a second order homogeneous linear differential equation whose general solution is A tan x + B sin x (A, B constant). [Hint: Use the fact that tan x and sin x are, individually, solutions and solve for the coefficients in standard form.]

Answer 1

Let y(x) = A tanke + B sinx ya dy da A sec²x + 3 cosa and d² y"= x2 - A ( 2 secx) (secr toun) - B sina Adding (1) and (3) we get d²4 + 4 = A (2 sec'a +i) tuna da? A - (y"+y) tana (2 sec?xt and then subtracting Multiply eqn (2) by , 2 tana from (3) we get y" - (2 tannly' = 2A secar. tunni - 13 sinx 2A sextana 2B tana.cosa Y"- (2 tanay! -B sina - 2Bsina = -3B sina :: B- y"_29' tana -3 sina zyotanx-y" 3 sinn anx -y") sinn By (1), (4) and (5), we get yo y"+y xtanat (2y't anx taxu (25ec²e +1) 3 sign y = y"+y 2y' tanu.sinne y". sinn asec²xti 3 sina 3 sign. +

Y = 4"ty + 3 y' tann (2 secx+1) - " :. 3(25ec’eti) y = 3(y"+y) + 2 tann ( 2 sec?se+1) 4' -Y"(25ec2x+1) 3y"+ 3y + 2 tanx ( 2sec+x+1)y' - (25ec 2e +1) y" - 3 (zsecse +420 (3 - 2secr-19"+ 2tanx ( 2sec že +1)y'- 3(2 sec?ze +1)y=0 -2 (sec's-1)y''+ 2tana ( 2 sec*x+1) y - 3(25ec3e+1)4 = 0 2 tanse.y" + 2 tana ( 2sec še +iJy'- 3(2sec še +134= 0