21復 旝暘曽掱幃(扨撈曽掱幃)

1.旝暘曽掱幃(扨撈曽掱幃)
x幉忋傪堦掕偺懍搙侾偱摦偔摦揰P偺帪崗t偵偍偗傞埵抲(x嵗昗)傪x(t)偲昞偡偲丄x'(t)=1偑惉傝棫偪傑偡丅 x'(t)=1傪枮偨偡娭悢x(t)傪媮傔偰傒傑偟傚偆丅

>dsolve(diff(x(t),t)=1,x(t))


摿偵丄弶婜忦審 x(0)=0傪枮偨偡娭悢傪媮傔傞偲偒偵偼丄

>dsolve({diff(x(t),t)=1,x(0)=0},x(t))


偲偟傑偡丅傑偨丄x幉忋傪堦掕偺壛懍搙-1偱塣摦偡傞摦揰P偺帪崗t偵偍偗傞埵抲(x嵗昗乯傪 x(t)偲昞偡偲丄x''(t)=-1偑惉傝棫偪傑偡丅x(t)傪媮傔傞偵偼師偺傛偆偵偟傑偡丅

>dsolve(diff(x(t),t,t)=-1,x(t))丂
(旝暘曽掱幃丂x''(t)=-1傪夝偔)

>dsolove({diff(x(t),t,t)=-1,x(0)=0,D(x)(0)=1},x(t))丂
(旝暘曽掱幃x''(t)=-1傪弶婜忦審x(0)=0,x'(0)=1偺壓偱夝偔)

拲堄丂旝暘曽掱幃傪夝偔偵偼丄dsolve(旝暘曽掱幃,x(t)) 偁傞偄偼dsolve({旝暘曽掱幃丄弶婜忦審},x(t))側偳偲擖椡偟傑偡丅

幚廗21.1丂師偺旝暘曽掱幃偺弶婜抣栤戣傪夝偒側偝偄丅
(1)v'(t)=-g,v(0)=v0丂丂丂(2)x''(t)=-g,x(0)=x0,x'(0)=v0

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俀. 憹怋偲尭悐
僶僋僥儕傾偑梕婍偵擖偭偰偄偰丄堦掕偺憹怋棪k偱憹怋偡傞偲偟傑偡丅偙偺偲偒丄帪崗t偵偍偗傞僶僋僥儕傾偺検傪x(t)偱昞偡偲丄
x'=kx丂(x'(t)=kx(t))


偑惉傝棫偪傑偡丅傑偨憹怋棪偑k-x,(k-x)(x-兩)偱偁傞偲偒偵偼丄

x'=(k-x)x, x'=(k-x)(x-兩)x


偑偦傟偧傟惉傝棫偪傑偡丅僷僢働乕僕DEtools偺DEplot僐儅儞僪傪梡偄傞偲丄偙傟傜偺旝暘曽掱幃偺夝嬋慄傪恾帵偡傞偙偲偑偱偒傑偡丅

>with(DEtools):
>de1:=diff(x(t),t)=x(t)
(旝暘曽掱幃x'=x傪de1偲偍偔)

>DEplot(de1,x(t),t=-5..5,x=0..2)丂
(旝暘曽掱幃de1偺夝嬋慄傪-5亝t亝5,0亝x亝2偺斖埻偵昞帵)

>init1:={[0,0],[0,0.5],[0,1]}
(弶婜忦審x(0)=0偲x(0)=0.5偲1傪巜掕)

>DEplot(de1,x(t),t=-5..5,x=0..2,init1)
(旝暘曽掱幃de1偺弶婜忦審init1傪枮偨偡夝嬋慄傪昞帵)


幚廗21.2丂師偺旝暘曽掱幃偺弶婜抣栤戣偺夝嬋慄傪昞帵偣傛丅
丂(1)[儘僕僗僥傿僢僋曽掱幃]x'=(1-x)x,弶婜忦審丂x(0)=0, 0.3, 0.7, 1.0
丂(2)x'(t)=(1+x)x丂弶婜忦審丂x(0)=0, 0.7, 1.0
丂(3)x'(t)=(1-x)(x-0.5)x 弶婜忦審 x(0)=0, 0.2 ,0.4, 0.6, 0.8, 1.01

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幚廗21.3[敿尭婜]
x'=-kx,x(0)=a傪枮偨偡娭悢x(t)偑x(t)=a/2偲側傞偺偼丄t=(log2)/k偺偲偒偱偁傞偙偲傪帵偟側偝偄丅丂

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3.怳摦
扨怳摦傪婰弎偡傞旝暘曽掱幃x''=-x偺夝傪媮傔丄偦偺僌儔僼傪昞帵偟偰傒傑偟傚偆丅

>dsolove(diff(x(t),t,t)=-x(t),x(t))

>dsolve({diff(x(t),t,t)=-x(t),x(0)=1,D(x)(0)=0},x(t)))

>plot(rhs(%),t=0..50)
(捈慜偺寢壥(%)偺塃曈偺僌儔僼傪昞帵)


幚廗21.4丂[杸嶤傪敽偆怳摦(尭戅怳摦)]
丂(1)x''=-x-kx'偺夝傪k=0.2, 2, 20偺応崌偵媮傔側偝偄丅
丂(2)x''=-x-kx',x(0)=1,x'(0)=-2偺夝傪k=0.2, 2, 20偺応崌偵媮傔丄偦傟傜偺僌儔僼傪昞帵偣傛丅

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幚廗21.5丂[嫮惂怳摦傪敽偆怳摦]
丂x''=-x+sin冎t,x(0)=0,x'(0)=0 偺夝傪冎=0, 0.2, 0.5, 0.9, 1, 1.5偺応崌偵媮傔丄偦傟傜偺僌儔僼傪0亝t亝500偺斖埻偱昞帵偣傛丅

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