Find the general solution of the given higher-order differential equation.d3u / dt3 + d2u / dt2 – 2u = 0 u(t)=

## Answer

Solve 16 due + 40 d’y(O + 25 y(x) = 0 Assume a solution will be proportional to e* for some constant A Substitute y(x)-c* into the differential equation Substitute (e’*) = ג 4er and (ex) = λ2 Factor out e* Since ex λ 0 for any finite λ, the zeros must come from the polynomial 16λ4 +40 λ2 + 25 = 0 Factor: 422+5)0 Solve forA The roots λ = ± NS both have muliplicity 2 and give yı(x) = ci e2ìV5x as solutions, where c1, c2, c3, and c4 are arbitrary constants The general solution is the sum of the above solutions: C1 e2 e 2 Apply Euler’s identity eatiß-ea cos() + i ea sin(β) Regroup terms (c 1 + C2 ) COS|-| + (c3 + C4 ) X COS Redefine cut c2 as c1-1 (cl-c2) as c2, сз + c4 as c3, and i (C3-c4) as c4 , since these are arbitrary constants Answer y(x) c2 sin

dx dx Assume a solution will be proportional to e1x for some constant λ Substitute y(x)-c* into the differential equation Substitute ī (ex)-יג e1x and dr (e’ 1-12 Factor out e* Since en 0 for any finite λ, the zeros must come from the polynomial Factor: a-1)(a2 +21+2-0 Solve for λ as solutions, where c1 and c2 are arbitrary constants The root 1 gives y3(x)-c3 ex as a solution, where c3 is an arbitrary constant. The general solution is the sum of the above solutions: y(x) = y1 (x) +Y2(x) + y3 (x) =ー-+- C2 ell-i)x Apply Euler’s identity ea+p-ea cos(B) + i ea sin(β) y(x) = c3 er + Ci (CoslX) + i sin(x))to( cos(x) i sin(x) ) Regroup terms x i(c1 -c2) sin(x) (C1 c2) cos(x) Redefine c c2 as c1 and i (c1 -C2) as c2 , since these are arbitrary constants Answer xC2 sin(x)c1 cos(x)

Consi der the following differential equati on du du dt dt Rewrite the above equation as follows: First write auxillary equation and then find the roots of the auxill ary equation auxillary equati on is f(m)-0 m3 +2m2-m2 +2m-2m-20 m(㎡ + 2m + 2)-1(ma +2m + 2)=0 (m-1)( +2m+2) 0 The roots are m =1,-1+i,-1-i The general solution is as follows )cossin) Therefore, the general solution of higher differenti al equati on is