设f(x)=x+∫(0,x)f(u)du ,f(x)是可微函数,求f(x)
绛旓細璁綟(x)=鈭玔0,x]f(t)dt 鍒橣(-x)=鈭玔0,-x]f(t)dt 浣滄崲鍏僽=-t,鍒檇t=-du,褰搕浠0鍙樺埌-x鏃,u浠0鍙樺埌x F(-x)=鈭玔0,x]f(-u)*(-du)=-鈭玔0,x]f(u)du =-F(x)鈭碏(x)涓哄鍑芥暟
绛旓細=G1(x)鑻f(x)涓哄伓鍑芥暟,鈭紙0,x锛塮(t)dt璁颁綔G2(x)G2(-x) = 鈭紙0,-x锛塮(t)dt = 鈭紙0,x锛塮(-u)d(-u)= -鈭紙0,x锛塮(u)d(u)= -鈭紙0,x锛塮(t)dt =G2(x)
绛旓細瑙ｏ細鈭玣'(x)dx=鈭玿dx=0.5x²+C1 娉ㄦ剰鍒f(0)=0,鎵浠1=0 鎵浠モ埆f(x)dx=x³/6 +C2
绛旓細瀵瑰凡鐭ュ紡姹傚寰梖'(x)=e^x+f(x),璁緔=f(x),寰 y'-y=e^x,鈶 鐢眣'-y=0寰梱=ce^x 璁緔=c(x)*e^(x),鍒檡'=[c'(x)+c(x)]e^x 浠ｅ叆鈶狅紝c'(x)=1 c(x)=x+c,鈭f(x)=(x+c)e^x 浠ｅ叆宸茬煡寮忥紝(x+c)e^x=e^x+鈭<0,x>[(t+c)e^t]dt =e^x+锛坸+c-1锛塭^...
绛旓細鍋囪f(x0鐨勪竴涓師鍑芥暟鏄F(x)鍒欏師寮=F(x)-F(0)
绛旓細鏈嬪弸锛屾偍濂斤紒涔变竷鍏碂绛旀鐪熷锛佽缁嗗畬鏁存竻鏅拌繃绋媟t锛屽笇鏈涜兘甯埌浣犺В鍐抽棶棰
绛旓細[F(x)]'=[鈭玔0,x] sint/t dt ]'=sinx/x 涓鑸舰寮忕殑銆愬彉鍔ㄤ笂闄愮Н鍒嗘眰瀵兼硶鍒欍戜负锛氥愨埆[蠁(x) ,蠄(x)] f(t)dt銆' = f(蠁(x))蠁'(x)-f(蠄(x))蠄'(x)璁惧嚱鏁皔=f(x) 鍦ㄥ尯闂碵a锛宐]涓婂彲绉紝瀵逛换鎰弜鈭圼a锛宐]锛寉=f(x)鍦╗a锛寈] 涓婂彲绉紝涓斿畠鐨勫间笌x鏋勬垚涓绉嶅搴...
绛旓細璇佹槑锛氬仛鍙橀噺鏇挎崲锛屼护x-t=s锛屽垯t=x-s,dt=-ds锛屼唬鍏モ埆锛0鈫x)f(t)g(x-t)dt寰楀埌鈭紙0鈫抶)f(t)g(x-t)dt=鈭紙x鈫0)f(x-s)g(s)(-ds)=鈭紙0鈫抶)f(x-s)g(s)ds=鈭紙0鈫抶)g(t)f(x-t)dt (鍥犱负瀹氱Н鍒嗕笌鎵閲囩敤鐨勭鍙锋棤鍏筹級銆傚鏋滃嚱鏁皔=f(x)鍦ㄧ偣x=x0澶勫強鍏堕檮杩戞湁...
绛旓細杩欎釜绉垎涓嶅垵绛夛紝涓嶈繃鍙互鐢ㄦ棤绌风骇鏁拌〃绀哄嚭鏉ョ殑 鍏朵腑绉垎閮ㄥ垎鐨勮绠楀涓 鐢ㄨ蒋浠堕獙绠楄繃鏁板硷紝涔熸槸鐩哥瓑鐨
绛旓細[鈭Н鍒嗕笂闄愬嚱鏁帮紙x锛0锛塮锛坹锛塢'=x鈥*f锛坸锛=f锛坸锛夊皢鍘熷紡灞曞紑,鐢变簬鏄t鐨勭Н鍒,锛坸-t锛変腑鐨剎鏄父鏁,鍙互鎻愬嚭鏉モ埆(0,x) (x-t)f(t)dt = x鈭(0,x) f(t)dt - 鈭(0,x) t f(t)dt 瀵箈姹傚寰 鈭(0,x) f(t)dt + xf(x) - xf(x) = 鈭(0,x) f(t)dt銆
