Question

Solve the differential equation by variation of parameters. Y"' + 3y' + 2y = 6 > 9+ et

Answer 1

Sol? 9" + 3y + 2y = 4+ex (D*+ 30+2)} = gtext , where Daten Auseilasy equation is per a (m+i) (m+2) = 0 => m= -1,-2 Complementary solution (say še) is Tyc = cek + ge-2x, where c, fcq are arbitrary constants, 1 m2 + 3 m+ 2=0

{ variation of parameter: Consider the differential equation y" + 90) y' + 8(x) y = g(5), whese 9682, 8(be) and goes are functions of x. Assume that y, (B) and yala) are a fundamental set of solutions for y" + 960) y' + 8(x)} =0 Then a particular solution to the non homogeneous differen- tial equation is yo = -y, l ya g(x) dx + yg s Gigase) dx where W( 3,42) = Y, Já - Yg y' should be non-zero, } 2 non-

. Now, Let = 8-23 us assume ume Y, (%) = e-x, Ya(x) = 6-2. So, y'() = -eX & ' (a) = -26-2* W($1,da) = (2+*)(-24-28) - (-20) (-e-*) W(2, 2) = -2e - 3* + e-3x = -2-3x to {e-3x = of нем, асх) 2 бет - Particulas solution (say yp) is 8p= -cc* J *.dk + e-** *** Hotek olx (yp= ex fatore dx - e-2x ex re dx - X Yp=-e-2/ e-2x YteX e da te-2x / e -da J -e-32

Now, put ate?=m, then el dx = dm . So, station en pel te} Se hele ak = d m = Inpol They Sen | 94e447 ex iex 9tex . put 9+et=n, then e* dx = dn & eX=n-9. So, ere dx = forma de = f(-2) dn = jan - Je don = n-9 los/n/ s slidx = x+c ļ is do y = lon/olte lear roue dx = e*+9 - 9 ln / e* + 911 So, Jp = e-* In 19+ (*) - 1-2x { e* +9 -9 lon / e%+91) - Jp = e-* In 19+e" - e-n_e-2x 9 + 9 1-2x lon / e%+91 222 9tea

So, the general solution of given differential equation (say ) is y = Ye + IP y = ge-* + G 1-2x + (9 1-2x + (-x) In/e%+91- e->_9e-2x = 6,-1)e* +(829)e«8* + (pe* + e**) (l/e*+91) put c-1 = ¢ & CG-9 = Ch' then Ty= c'e-t + g'e-2x + (9 1-2x + e-*) (len 10%+91) where ed ca are arbitsery constants. Ans.