Question

can you help me with 37, 41, and 45 thanks

3.6. Homogeneous Equations with Constant Coefficients 191 31. d(isin(2)) 34. 읊 (e212. cos(3x1+ sin x) 36. fe(l-2i)dx 32. (e3iz) 33. ,e* sin(z)) 37. Je(1+2)dr In Problems 39- 42, solve the system of equations for the possibly complex- valued unknowns 39. iAi-1, A1 iA2 0 40. iA21, 2A1 iA20 43. Show that eir is a solution to x" +x 0 44. Show that ei2z is a solution to z', + 4x 0 45. Use separation of variables to show that e(a+ib)t Is a solution to r, = (a +ib)r

Answer 1

integral e^(1 + 2i)x dx Let I = integral e^(1 + 2i)x dx rightarrow (1) I = integral e^x middot e^2in dx I = integral e^2ix_u e^x_v middot dx rightarrow (2) integral uv = u integral v - integral u integral v I = e^2ix integral e^x - integral d (e^2ix) middot integral e^x dx d (e^2x) = 2e^2x I = e^2ix (e^x) - integral 2i e^2ix e^x dx I = e^2ix (e^x) - 2i integral e^2ix middot e^x dx I = e^2ix middot e^x - 2i integral e^2ix middot e^x dx I = e^2ix middot e^x - 2i (I) (by (2)) I + 2i (I) = e^2ix middot e^x I [1 + 2i] = e^2ix middot e^x I = e^2ix middot e^x/1 + 2i I = e^(1 + 2i)x/1 + 2i \r\n A_1 - iA_2 = 1 rightarrow (1) 2A_1 + iA_2 = i rightarrow (2) rightarrow (1) times 2 2 A_1 - 2i A_2 = 2 2 A_1 + iA_2 = i - 3 iA_2 = 2 - i A_2 = 2 - i/- 3i [i^2 = -1] A_2 = -2 (2 - i)/3i times 3i/3i = -(2 - i) 3i/9