Question

could someone help me with this

Solve the following initial value problem 22 y (z) – 7 (de v(z)) + 10 y (2) = 18 sin (z) – 14 cos (2) with d. 2 y(0)=2, dy(0) dr =12

Answer 1

consider the given IVP y"-74'tloy = 18 sina_14 asa with yo)=2 and y'co)=12 28 5x the characteristic equation is mm +10=0 » m=2,5 The complementary solution is 9c = Cј е + Ce Let the particular Solution be yp = A CSX + B Sinx then yp = - Asinx +BC8x and yp A csr – B sina substituting Sp, Up and up in the given DE - A COX-B Sine - 7 (-A5nX+803x) +10 (A CBX+B 5.0%) = 18 sins – 14 Cosa (-A-78 +10A) CoDX +(-8 + 7A +106)S 18 Sinx -1468X b → 9A-7B) cosa +(9B+7A) sind = 18 Sinx -14087 By equating we get

9 A-7B = -14 and 7A+ 9B = 18 solving these equations, we get Azo and B=2 The particular solution is up=asina The general solution is y=yc typ 2x y=ce + C2 e Then yco) = 2 -> at C₂ = 2 (). and y = 20 = 20 é + 5e +2082 j'co) = 12 » 20, +50₂ + 2 =12 +2sina > 20 + 50₂ = 10 (2) solving (1) and (2), we get a=0 and C2=2 The solution of the given Ivp is. ycal = 2e + 2C8a