Question

(A) Find the largest x-interval where the initial value problem has a unique solution:

$(x^{2}-4x)y^{(6)}+\frac{1}{ln(x-1)}y^{(5)}+(x+1)y'''+e^{x}y' +(tanx)y = \frac{1}{x^{2}-9}; y(2.5)=A, y'(2.5)=B,y''(2.5)=C,y'''(2.5)=D,y^{(4)}(2.5)=E,y^{(5)}(2.5)=F,$

Where A, B, C, D, E, F are some known constants.

(B) Determine whether the set of functions ${5,sin^{2}x,cos2x}$ could form a fundamental set of solution of a linear differential equation

Thank you

Answer 1

We first state a hypothesis Hypothesis to consider the nth-order lineer differential equation Aota) dan tin Amlosg = F2-0 7 where ao, a, an and F a on asneb. dan tanud (2) dy + are continuoue real functione real interval acash and 9o (2) to for any be any point of the interval asasb, and arbitrary real contente 2. Lut no be n Co, C... Cnrs Conclusion: There a fr-1/2) = Court exists unique solution of of 1) such that floo) = Co, f'(ao) = (1, ..., and the solution is defined over the entire interval alacb. Here (5) 1 + (241) y" + ea y + (tenn) y = + (22_42) y tn (2-1) > ao (8) = a -4a; Az (n) = (2+1) a2(a)=0 ; In (2-1) A4 Bx)=0 ; a g(n) = e"; ac(a)=tana 1 ans Oo om So Noua a2 a3 a4 a5, ao all continuous on R. ay is Continuons on (1,0) and F. in continuous IR-{£3} . ao, a,... a and F are all continuous on (1, 3). A (2)=0 at 2004 and 0, 4 & (2, 3). 2.5 E (1,3). where the initial (43) is the largest a interval value problem has a unique solution. Also, Moreover Hence

*) let fica = 5 ; fzco) - Sinha; fz(n) = cos 22 . fila)= fi"(a) = 0; fal () = 2.Sina cos 2 = Sin27 ; fè" (m) =20052* , The wronokian of filfa and f3 ficas fa(2) faca) film) f(a) film) وا fólm) --2 Sin 27 folcm) =-400520 5 Sinra COS 24 11 2 Sina cosa -2 Sin27 0 0 2 Cos 20 -4 cos 22 CS 22 5 TV [- 8 Sina cosa. + 4 Sinza cos20] 25 [- 4 Sinza: Cos 20 + 4 Sinza Cos2a] so a we Know Solutions fi to and fa from fundamental set of of a linear differential equation if and and only if wronskian of fi fa, and f3 is never zero. wromskian of forfa and fo is zero for fa and fa could not form a fundamental set of solutions of a linear differential equation the AA