Question

Use any method to solve the problem: 4y" - y = ex/2 + 4,

Answer 1

to solve the thod problem me use any uy" - y +4. sol H2 4- y е +4. 0 y"=D²y, y'ady y" and yl values substistute inequation 212 tu. 40²9 -y- e x12 (402-1)= e +4. The homogeneous differential equation is (4D-Dy=0 Ausilary equation is 40²_1=0 (2D-1)(2D+1) =0 D = 1, 1 roots are - 1 and he The solution of the homogeneas differential equation- the complementary solution is the roots are of B. Ye (x) = ctx + cq eBx os 8 = 1 2

y(x) = geta x + ez Yele) = Gette emne particular integral (p. I). Ypres) x12 +4: (4D2-1) p = 지오 + Ypa 14.) 4D²-1 402-1 ( annihilator methodo particulas solution. ax αα e it fra) to ypa (CD) fra) - aix 2 ax it f(a)=0 f(a) fla +0 ala y pa +4 4D²4 You 40²-1 L hede D = 1 hele b=0 *12 12.4 at u Yorama 464²-) You SO V 개요 y prxda e 8 D here D = £ x e242 -4 = 8(5) xel Yp(x)= -4 212_4 Jp (x)= xe xo 16 4.

216 xe Yp (x)= -4. 4. ५ particular solution pro particular integral is a ez Oprava 4 4 The general Solution is sum of complementary solution & particular Integral y(x) Yerx) + 4pca) y(x)= ce Etape + 28 xez -4. 4 -2 2 х xez ta 1/(x)=ce 16 4