32 TT Q6 (20 points): a) Classify the following DE: x?y”+ay/4+3y/16- |-7 x=0. b) Show that: yi(1) =.

32 TT Q6 (20 points): a) Classify the following DE: x?y”+ay/4+3y/16- |-7 x=0. b) Show that: yi(1) = VTx cos(Vx)and yz(x)=1Vx sin (Vx) are linearly-independent functions. c) By solving the DE as Euler-Cauchy equation; show that y(x) and y, (x) are fundamental solutions of homogeneous form of the given DE; d) Find the particular solution of the given non-homogeneous DE in terms of the cosine and sine Fresnel functions (C(x), S(x)), which are defined, respectively, as follows: C(x)= [cos (12) , S(x)= { sin ( 12.) a 0 e) Use the transformation: y=uVx and z=Vx to convert the non-homogenous form

32 TT Q6 (20 points): a) Classify the following DE: x?y”+ay/4+3y/16- |-7 x=0. b) Show that: yi(1) = VTx cos(Vx)and yz(x)=1Vx sin (Vx) are linearly-independent functions. c) By solving the DE as Euler-Cauchy equation; show that y(x) and y, (x) are fundamental solutions of homogeneous form of the given DE; d) Find the particular solution of the given non-homogeneous DE in terms of the cosine and sine Fresnel functions (C(x), S(x)), which are defined, respectively, as follows: C(x)= [cos (12) , S(x)= { sin ( 12.) a 0 e) Use the transformation: y=uVx and z=Vx to convert the non-homogenous form of the given DE into a Bessel’s DE. f) Solve the resultant DE in terms of Bessel’s functions. g) Find relations between the solutions of Part (c) and Part (f), show details. h) If the error function is defined by: erf(3) = du show that the error function is related to Fresnel functions by the following relation: erfidia 12 (C(x)=is(x)) where i=V=1

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