%7VO H.W.pdf Erereises. Solve the following differential equations: (1) "-- by =0 (2) "+ 2y =0 (3) " + 2y +y =0 (4) " + 9y = 0 (5) " - G/ + 13y = 0 (6) " - 4 + 5y = 0 (7) "- 4y + 4y = 0 (8) " + 6y +...

حل سؤال 19

Extracted text: %7VO H.W.pdf Erereises. Solve the following differential equations: (1) "-- by =0 (2) "+ 2y =0 (3) " + 2y +y =0 (4) " + 9y = 0 (5) " - G/ + 13y = 0 (6) " - 4 + 5y = 0 (7) "- 4y + 4y = 0 (8) " + 6y + 10y = 0 (9) " + -0 (10) "+ 6y + 13y = 0 (11) 10y" - 3y -y=0 (12) 4" + 4y + 10y =0 Solve the initial value problems: (13) y" - 4y +3y = 0; y(0) - -1, / (0) = 3 (14) y" + 4y= 0; y() 1, () = -4 (15) 6y" - - y = 0; y(0) = 10, g'(0) = 0 13 (16) 4" - 4y - 3y- 0; v(0) = 12(0) - (17) y"+ 7y' + 12y = 0; y(0) = -1,(0) = 0 (18) 36y" – 12y + y = 0; y(0) = 3, (0) = (19) (a) Suppsoe that y is a solution of the differential equation ay" + by + cy = 0. Let :(r) - s(r - ro) where ro is an arbitrary real number. Show that a:" + be' + es =0 (b) Let z1(a) = yn(r- ro) and z2(x) = 2(r - ro), where yG = CI1 + caa is the general solution to ay" + by +cy = 0. Show that se = e +gz is also a general solution of ay" + by/ + cy =0.