\\(f(x)\\) \u3092\u6574\u5f0f\u3067\u8868\u3055\u308c\u308b\u95a2\u6570\u3068\u3057, \\(g(x) = \\displaystyle\\int _ 0^x e^t f(t) \\, dt\\) \u3068\u304a\u304f.\r\n\u4efb\u610f\u306e\u5b9f\u6570 \\(x\\) \u306b\u3064\u3044\u3066\r\n\\[\r\nx \\left( f(x) -1 \\right) = 2 \\displaystyle\\int _ 0^x e^{-t} g(t) \\, dt\r\n\\]\r\n\u304c\u6210\u308a\u7acb\u3064\u3068\u3059\u308b.<\/p>\r\n
(1)<\/strong>\u3000\\(x f''(x) +(x+2) f'(x) -f(x) = 1\\) \u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(f(x)\\) \u306f\u5b9a\u6570\u307e\u305f\u306f \\(1\\) \u6b21\u5f0f\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n (3)<\/strong>\u3000\\(f(x)\\) \u304a\u3088\u3073 \\(g(x)\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\[\r\nx \\left( f(x) -1 \\right) = 2 \\displaystyle\\int _ 0^x e^{-t} g(t) \\, dt \\quad ... [\\text{A}]\r\n\\]\r\n\u6761\u4ef6\u3088\u308a\r\n\\[\r\ng'(x) = e^x f(x) \\quad ... [1]\r\n\\]\r\n[A] \u306e\u5de6\u8fba, \u53f3\u8fba\u3092\u305d\u308c\u305e\u308c \\(u(x) , v(x)\\) \u3068\u304a\u304f.\r\n\\[\\begin{align}\r\nu'(x) & = x f'(x) +f(x) -1 , \\\\\r\nv'(x) & = 2 e^{-x} g(x) \\\\\r\n\\text{\u2234} \\quad 2 e^{-x} g(x) & = x f'(x) +f(x) -1 \\quad ... [2]\r\n\\end{align}\\]\r\n\u3055\u3089\u306b\r\n\\[\\begin{align}\r\nu''(x) & = f'(x) +xf''(x) +f'(x) = x f''(x) +2f'(x) , \\\\\r\nv''(x) & = -2e^{-x} g(x) +2e^{-x} \\cdot g'(x) \\\\\r\n& = -\\left( x f'(x) +f(x) -1 \\right) +2 f(x) \\quad ( \\ \\text{\u2235} \\ [1] [2] \\ ) \\\\\r\n& = -xf'(x) +f(x) +1 \\\\\r\n\\text{\u2234} \\quad x f''(x) & +2f'(x) = -xf'(x) +f(x) +1 \\\\\r\n\\text{\u2234} \\quad x f''(x) & +(x+2) f'(x) -f(x) =1\r\n\\end{align}\\]\r\n (2)<\/strong><\/p>\r\n \\(f(x)\\) \u3092 \\(x\\) \u306e \\(n\\) \u6b21\u5f0f\u3068\u3057, \\(n\\) \u6b21\u306e\u4fc2\u6570\u3092 \\(a \\ ( a \\neq 0 )\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\nf(x) & = a x^n + \\cdots , \\ f'(x) = na x^{n-1}+ \\cdots , \\\\\r\nf''(x) & = an(n-1) x^{n-2} + \\cdots\r\n\\end{align}\\]\r\n (1)<\/strong> \u306e\u7d50\u679c\u306e\u5de6\u8fba\u306b\u7740\u76ee\u3059\u308b\u3068, \u5404\u9805\u306e\u6b21\u6570\u306f\r\n\\[\r\nxf''(x) : \\ n-1 \\text{\u6b21}, \\ (x+2) f'(x) : \\ n \\text{\u6b21} , \\ f(x) : \\ n \\text{\u6b21}\r\n\\]\r\n\u306a\u306e\u3067, \u5de6\u8fba\u306e\u6b21\u6570\u306f\u9ad8\u3005 \\(n\\) \u6b21\u3067, \u305d\u306e\u4fc2\u6570\u306b\u7740\u76ee\u3059\u308b\u3068\r\n\\[\\begin{align}\r\nx \\left( an x^{n-1} \\right) & -\\left( a x^n + \\cdots \\right) \\\\\r\n& = a(n-1) x^n + \\cdots\r\n\\end{align}\\]\r\n\\(n \\geqq 2\\) \u3068\u3059\u308b\u3068, \u5de6\u8fba\u306f \\(n\\) \u6b21\u3068\u306a\u308a, \u53f3\u8fba\u304c\u5b9a\u6570\u3067\u3042\u308b\u3053\u3068\u306b\u77db\u76fe\u3059\u308b.\r\n\u4e00\u65b9<\/p>\r\n \\(n=1\\) \u306e\u3068\u304d, \u5de6\u8fba\u306f \\(1\\) \u6b21\u306e\u9805\u304c\u6d88\u3048\u3066, \u5b9a\u6570\u9805\u3068\u306a\u308b.<\/p><\/li>\r\n \\(n=0\\) \u306e\u3068\u304d, \u5de6\u8fba\u306f\u5b9a\u6570\u9805\u3068\u306a\u308b.<\/p><\/li>\r\n<\/ul>\r\n \u3088\u3063\u3066, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n (3)<\/strong><\/p>\r\n (2)<\/strong> \u306e\u7d50\u679c\u3088\u308a, \\(f(x) = ax+b\\) \u3068\u304a\u3051\u308b.
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
\r\n
\r\n(1)<\/strong> \u306e\u7d50\u679c\u3092\u7528\u3044\u308c\u3070, \\(f'(x) = a\\) , \\(f''(x) = 0\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\na(x+2) -ax-b & = 1 \\\\\r\n\\text{\u2234} \\quad b & = 2a-1 \\quad ... [3]\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\ng(x) & = \\displaystyle\\int _ 0^x ( at +2a-1 ) e^t \\, dt \\\\\r\n& = a\\big[ t e^{t} \\big] _ 0^x -a \\displaystyle\\int _ 0^x e^t \\, dt +(2a-1) \\displaystyle\\int _ 0^x e^t \\, dt \\\\\r\n& = ax e^x -(a-1) \\big[ e^t \\big] _ 0^x \\\\\r\n& = ax e^x -(a-1) \\left( e^x -1 \\right) \\quad ... [4]\r\n\\end{align}\\]\r\n\u3055\u3089\u306b\r\n\\[\\begin{align}\r\nv(x) & = 2 \\displaystyle\\int _ 0^x e^{-t} \\left\\{ at e^t -(a-1) \\left( e^t -1 \\right) \\right\\} \\, dt \\\\\r\n& = 2 \\displaystyle\\int _ 0^x \\left\\{ at -(a-1) \\left( 1-e^{-t} \\right) \\right\\} \\, dt \\\\\r\n& = 2 \\left[ \\dfrac{at^2}{2} -(a-1) \\left( t +e^{-t} \\right) \\right] _ 0^x \\\\\r\n& = ax^2 +2(a-1) \\left( x +e^{-x} -1 \\right)\r\n\\end{align}\\]\r\n\u3053\u3053\u3067\r\n\\[\r\nu(x) = x \\{ ax +2(a-1) \\} = ax^2 +2(a-1)x\r\n\\]\r\n\u306a\u306e\u3067, [A] \u3088\u308a\r\n\\[\r\n2(a-1) \\left( e^{-x} -1 \\right) = 0\r\n\\]\r\n\u3053\u308c\u304c\u4efb\u610f\u306e\u5b9f\u6570 \\(x\\) \u3067\u6210\u7acb\u3059\u308b\u306e\u3067\r\n\\[\\begin{align}\r\na-1 & = 0 \\\\\r\n\\text{\u2234} \\quad a & = 1\r\n\\end{align}\\]\r\n[3] \u3088\u308a, \\(b =2-1 =1\\) .
\r\n\u3088\u3063\u3066, [4] \u3088\u308a\r\n\\[\r\nf(x) = \\underline{x+1} , \\ g(x) = \\underline{x e^x}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(f(x)\\) \u3092\u6574\u5f0f\u3067\u8868\u3055\u308c\u308b\u95a2\u6570\u3068\u3057, \\(g(x) = \\displaystyle\\int _ 0^x e^t f(t) \\, dt\\) \u3068\u304a\u304f. \u4efb\u610f\u306e\u5b9f\u6570 \\(x\\) \u306b\u3064\u3044\u3066 \\[ x \\left( f(x … \u7d9a\u304d\u3092\u8aad\u3080