Question

Problem 3: Find a solution to the IVP dy dy + dc2 + y = 0, y(0) = y'(0) = 1. dx Problem 4: Suppose you are given the differential equation ay" +by' + cy = 9(2) where a, b, and c are constants. For each of the following choices of g(x), write down the form for the particular solution Yp that you would use: (a) g(x) = 205 (b) g(x) = x²e32 (c) g(x) = xº cos(x) (d) g(x) = cos(x) + cos(2x) + cos(3.c)

Answer 1

to the lup + x2ه 0 find the solution dy Ans - Given dy dy + dx² de This is im homogo neous est 11 d dt est ert zo tre dx+y=0, 7(0) = y'(0) </ 4 y=0, 3/c := y(0) = 1, g(0) = the foom of second ooder linear ODE has the form of ay"+by't cy=0 Rewrite the equatim y = e ។ (1004) alveot) = 0 (0ot) como y"+y'+ y = 0 . de (8²%&+1) -)0'+8+1=0 -bt√6²400 for quadratic formula 18,2= - + VC)2-4090) 2(1) - +/+1-4 -7,7-3--23) 74 53i for two complex real roots o, to where 8 = d+iB, 2 =-iß, the genesial Solution takes YzeQd (c,cos (pt) +62 Sim (pt)) -Źt & Zert feat 2 a 2 2 ご 2 the form Sim е е initial (c, cos( 13 t) + (2 condition yeol = 1 -t (6, cos (EX) + 6, Sim (4) Apply y=e

t= oy (o)=e Sm/13t 2 (0,109 ((-) + Sim +(y Sim (£ (0)), (GC+6260) 1 - 1001>9+) - (cos(@4) +(2.5m y's (9) Cette (109 (24) +6,00 (24) * (cos(84) + (24))+etla (v3 rule 189=f'page) ģim (24) +r3 c, cos(1) y = e put y' (0) = 1. dt 2 Sin <-:Apply produet -e 2 put y'ra - 19 y'o-e % (109(E6)+ ezsin(6) % (-vz Sim (196+15c2.cos(eca) 2 31 = -1713c2 2 313 -) Ca -> -)2 = -/+03 (2 - 3 = √3cz 3 x 3 33 13 F =) C2 = √3) (cos(TE + ) + s sin (184) put (=v3. -t/2 :y=e -t/2 (tas(E+) 763 sim (6+) . Y=e