\\(f _ 0 (x) = x e^x\\) \u3068\u3057\u3066, \u6b63\u306e\u6574\u6570 \\(n\\) \u306b\u5bfe\u3057\u3066,\r\n\\[\r\nf _ n (x) = \\displaystyle\\int _ {-x}^x f _ {n-1} (t) \\, dt +f' _ {n-1} (x)\r\n\\]\r\n\u306b\u3088\u308a\u5b9f\u6570 \\(x\\) \u306e\u95a2\u6570 \\(f _ n (x)\\) \u3092\u5b9a\u3081\u308b.<\/p>\r\n
(1)<\/strong>\u3000\\(f _ 1 (x)\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(g(x) = \\displaystyle\\int _ {-x}^x (at+b) e^t \\, dt\\) \u3068\u3059\u308b\u3068\u304d, \u5b9a\u7a4d\u5206 \\(\\displaystyle\\int _ {-c}^c g(x) \\, dx\\) \u3092\u6c42\u3081\u3088. \u305f\u3060\u3057, \u5b9f\u6570 \\(a , b , c\\) \u306f\u5b9a\u6570\u3068\u3059\u308b.<\/p><\/li>\r\n (3)<\/strong>\u3000\u6b63\u306e\u6574\u6570 \\(n\\) \u306b\u5bfe\u3057\u3066, \\(f _ {2n} (x)\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\(F _ 0(x) = \\displaystyle\\int f _ 0(x) \\, dx\\) \u3068\u304a\u3051\u3070\r\n\\[\\begin{align}\r\nF _ 0(x) & = xe^x -\\displaystyle\\int e^x \\, dx \\\\\r\n& = (x-1) e^x +C \\quad ( C \\text{\u306f\u7a4d\u5206\u5b9a\u6570} )\n\\end{align}\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\nf _ 1(x) & = F _ 0(x) -F _ 0(-x) +e^x +xe^x \\\\\r\n& = (x-1)e^x +(x+1)e^{-x} +(x+1)e^x \\\\\r\n& =\\underline{2xe^x +(x+1)e^{-x}}\n\\end{align}\\]\r\n (2)<\/strong><\/p>\r\n \\[\\begin{align}\r\ng(-x) & = \\displaystyle\\int _ x^{-x} (at+b)e^t \\, dt \\\\\r\n& = -\\displaystyle\\int _ {-x}^x (at+b)e^t \\, dt = -g(x)\n\\end{align}\\]\r\n\u306a\u306e\u3067, \\(g(x)\\) \u306f\u5947\u95a2\u6570\u3060\u304b\u3089\r\n\\[\r\n\\displaystyle\\int _ {-c}^c g(x) \\, dx = \\underline{0}\n\\]\r\n (3)<\/strong><\/p>\r\n \u300c \\(f _ {2n}(x) = ( a _ n x+b _ n ) e^x\\) \uff08 \\(n\\) \u306f\u975e\u8ca0\u6574\u6570\uff09\u3068\u8868\u305b\u308b. \u300d ... [A] \u304c\u3059\u3079\u3066\u306e\u81ea\u7136\u6570 \\(n\\) \u306b\u3064\u3044\u3066\u6210\u308a\u7acb\u3064\u3053\u3068\u3092, \u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3092\u7528\u3044\u3066\u793a\u3059.<\/p>\r\n 1*<\/strong>\u3000\\(n = 0\\) \u306e\u3068\u304d\r\n\\[\r\na _ 0 = 1 , \\ b _ 0 = 0 \\quad ... [1]\n\\]\r\n\u3067, [A] \u304c\u6210\u7acb\u3057\u3066\u3044\u308b.<\/p><\/li>\r\n 2*<\/strong>\u3000\\(n = k \\ ( k \\geqq 0 )\\) \u306e\u3068\u304d 1*<\/strong> 2*<\/strong> \u3088\u308a, \u3059\u3079\u3066\u306e\u81ea\u7136\u6570 \\(n\\) \u306b\u3064\u3044\u3066, [A] \u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u304c\u793a\u3055\u308c\u305f.
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
\r\n
\r\n[A] \u304c\u6210\u7acb\u3057\u3066\u3044\u308b, \u3059\u306a\u308f\u3061, \\(f _ {2k} = ( a _ k x+b _ k ) e^x\\) \u3068\u8868\u305b\u308b\u3068\u4eee\u5b9a\u3059\u308b\u3068\r\n\\[\\begin{align}\r\nf _ {2k+2}(x) & = \\displaystyle\\int _ {-x}^x f _ {2k+1}(x) \\, dt +f' _ {2k+1}(x) \\\\\r\n& = \\displaystyle\\int _ {-x}^x \\left( \\displaystyle\\int _ {-t}^t f _ {2k}(s) \\, ds +f' _ {2k}(t) \\right) \\, dt \\\\\r\n& \\qquad +\\dfrac{d}{dx} \\left( \\displaystyle\\int _ {-x}^x f _ {2k}(t) \\, dt +f' _ {2k}(x) \\right) \\\\\r\n& = \\displaystyle\\int _ {-x}^x \\displaystyle\\int _ {-t}^t f _ {2k}(s) \\, ds \\, dt +f _ {2k}(x) -f _ {2k}(-x) \\\\\r\n& \\qquad +f _ {2k}(x) +f _ {2k}(-x) +f'' _ {2k}(x) \\\\\r\n& = \\underline{\\displaystyle\\int _ {-x}^x \\displaystyle\\int _ {-t}^t f _ {2k}(s) \\, ds \\, dt} _ {[2]} +2 f _ {2k}(x) +f'' _ {2k}(x)\n\\end{align}\\]\r\n(2)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\r\n[2] = 0\n\\]\r\n\u307e\u305f\r\n\\[\\begin{align}\r\nf' _ {2k}(x) & = a _ {k} e^x +( a _ k x +b _ k ) e^x \\\\\r\n& = ( a _ k x +a _ k +b _ k ) e^x , \\\\\r\nf'' _ {2k}(x) & = a _ {k} e^x +( a _ k x +a _ k +b _ k ) e^x \\\\\r\n& =( a _ k x +2a _ k +b _ k ) e^x\n\\end{align}\\]\r\n\u3086\u3048\u306b\r\n\\[\\begin{align}\r\nf _ {2k+2} & =2( a _ k x +b _ k ) e^x +( a _ k x +2a _ k +b _ k ) e^x \\\\\r\n& = ( 3a _ k x +2a _ k +3b _ k ) e^x\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\na _ {k+1} = 3a _ k , \\ b _ {k+1} = 2a _ k +3b _ k \\quad ... [2]\n\\]\r\n\u3068\u304a\u3051\u3070, \\(n = k+1\\) \u3067\u3082 [A] \u306f\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n<\/ol>\r\n
\r\n[1] [2] \u3088\u308a\r\n\\[\\begin{align}\r\na _ n & = 3^n , \\\\\r\nb _ {n+1} & = 2 \\cdot 3^n +3 b _ n \\\\\r\n\\text{\u2234} \\quad & \\dfrac{b _ {n+1}}{3^{n+1}} =\\dfrac{b _ n}{3^n} +\\dfrac{2}{3}\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\n\\dfrac{b _ n}{3^n} & = b _ 0 +\\dfrac{2n}{3} =\\dfrac{2n}{3} \\\\\r\n\\text{\u2234} \\quad b _ n & = 2 \\cdot 3^{n-1} n\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\nf _ {2n}(x) =\\underline{3^{n-1} ( 3x +2n ) e^x}\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(f _ 0 (x) = x e^x\\) \u3068\u3057\u3066, \u6b63\u306e\u6574\u6570 \\(n\\) \u306b\u5bfe\u3057\u3066, \\[ f _ n (x) = \\displaystyle\\int _ {-x}^x f _ {n-1} (t) \\, dt +f' … \u7d9a\u304d\u3092\u8aad\u3080