[Problem 1] (20 pts.)

a)[eqn: lambda ^{3} - lambda ^{2} - lambda +1=0`이`1(중근)과`-1`을`근으로`가지므로`일반해는# y=(a+bx)e ^{x} +ce ^{-x}# y(0)=a+c=2# y ^{prime } =(a+bx+b)e ^{x} -ce ^{-x} ,`y ^{prime } (0)=a+b-c=1# y ^{''} =(a+2b+bx)e ^{x} +ce ^{-x}# y ^{'} (0)=a+2b+c=0`로`부터# a=2,`b=-1`,`c=0# y=(2-x)e ^{x}]

b)

[Problem 2] (20 pts.)

a)homogeneous equation

[eqn: A= {bmatrix{1&4#1&1}} 의`고유치는`3,`-1# 각각의`고유벡터는` {pmatrix{2,`&1}} ^{T} ,` {pmatrix{2,`&-1}} ^{T}# y _{h1} =2c _{1} e ^{3t} +2c _{2} e ^{-t}# y _{h2} =c _{1} e ^{3t} -c _{2} e ^{-t}# # y _{p} =u`cost+`v`sint`로`두고`특수해를`찾으면# y _{p} = {bmatrix{cost-sint,`&0}} ^{T}]

b)[eqn: y= sum _{n=0} ^{INF } a _{n} x ^{n} ,`x ^{2} y= sum _{n=2} ^{INF } a _{n-2} x ^{n} ,# (4x ^{2} -2)y= sum _{n=2} ^{INF } (4a _{n-2} -2a _{n} )x ^{n} -2(a _{0} +a _{1} x),# -4xy ^{prime } =-4 sum _{n=0} ^{INF } na _{n} x ^{n} ,# y ^{'' } = sum _{n=0} ^{INF } (n+1)(n+2)a _{n+2} x ^{n} .# y ^{'' } -4xy ^{prime } +(4x ^{2} -2)y# =2a _{2} +6a _{3} x-4a _{1} x-2(a _{0} +a _{1} x)# + sum _{n=2} ^{INF } ((n+1)(n+2)a _{n+2} -4na _{n} +4a _{n-2} -2a _{n} )x ^{n}# y ^{'' } -4xy ^{prime } +(4x ^{2} -2)y=0이므로`a _{0} =a _{2} ,`a _{1} =a _{3} ,# a _{n+2} = {(4n+2)a _{n} -4a _{n-2}} over {(n+1)(n+2)} (n GEQ 2).# # 짝수항`;`a _{2k} =b _{k} 로`두면,`# b _{0} =b _{1} ,`b _{k+1} = {(8k+2)b _{k} -4b _{k-1}} over {(2k+1)(2k+2)} (k GEQ 1)로`부터#]

[eqn: b _{2} = {1} over {2} b _{0} ,`b _{3} = {1} over {3!} b _{0} ,`...`을`얻는다# 실제로`b _{k} = {1} over {k!} b _{0} `수열이`점화식을`만족한다# # 홀수항;`마찬가지로`a _{2k+1} =c _{k} 로`두면,# c _{0} =c _{1} ,`c _{k+1} = {(8k+6)c _{k} -4c _{k-1}} over {(2k+2)(2k+3)} (k GEQ 1)로`부터# c _{2} = {1} over {2} c _{0} ,`b _{3} = {1} over {3!} c _{0} ,`...을`얻는다# 그러므로`y=a _{0} (1+x ^{2} + {x ^{4}} over {2!} + {x ^{6}} over {3!} +` CDOTS )# `````````````````````````````````````````+a _{1} (x+x ^{3} + {x ^{5}} over {2!} + {x ^{7}} over {3!} + CDOTS )# =(a _{0} +a _{1} x)(1+x ^{2} + {x ^{4}} over {2!} + {x ^{6}} over {3!} + CDOTS )=(a _{0} +a _{1} x)e ^{x ^{2}}]

[Problem 3] (20 pts.)

a)

b)[eqn: n=5,`y=AJ _{5} (x)+BY _{5} (x)]

[Problem 4] (20 pts.)

b)

[eqn: sY _{1} -2+sY _{2} = {2} over {s ^{2} -1}# sY _{2} -1+sY _{3} = {1} over {s-1}# sY _{3} +sY _{1} -1= {2} over {s-1} + {1} over {s+1}# RARROW Y _{3} = {3} over {2} {1} over {s(s-1)} + {1} over {2s(s+1)} - {1} over {s(s ^{2} -1)}# = {2s} over {s(s ^{2} -1)} = {2} over {s ^{2} -1}# Y _{2} = {1} over {s} + {1} over {s(s-1)} - {2} over {s ^{2} -1} = {1} over {s+1}# Y _{1} = {2} over {s} - {1} over {s+1} + {2} over {s(s ^{2} -1)} = {1} over {s-1}# RARROW # y _{3} (t)=2sinh`t`,`y _{2} (t)=e ^{-t} `,`y _{1} (t)=e ^{t}]

[Problem 5] (10 pts.)

[Problem 6] (10 pts.)