Question

4. (20 points) (a) (15 points) Find the general solution of the linear differential equation eBay y" - 6y' +9y = on (0,-) using variation of parameters.

Answer 1

ex yl-64't gy = Ja soi W First tinding complementary solutions the Auxilary equation is m²6mtgo m²m-3 m +920 mim-3) -3 (m-3) = 0 (m-3) (n-3) 20 m=3,3; Yecx)=a 8 + G- Kare let y = 0 & J = x x 'y z then; wronksian of Y 4 Yaz y' Yal Wly,Y)= 3 x x 30 3x + 3x x wly, g) = 66 & + 3x fx_3 x 66 x wly Y ) = cox Now finding Particular solution using vanation of Coeftruants Ypala YJ Zz Jen) dat Ye Yi foxy.de Wlyy p3 WCY14) where tonia Ja

(geht H. =) Yos) = 4 asa ek + G x x Yprasa wet xet, old dx + x ² x Jer. esk.dk Ex Je Yor) = - BBx / x dx + x B x + dhe Ypin) = x (xt +xempo ht - 2 x/2.8x + 2X BX "2 3. on you a -2 a5x B + 2x te B 3 Now general solution is given by. YER) Yu + Yp (2) Apex) = -2/3 + Yara Box 3