微分方程式の話

暟椚㷕

痥㔐湡

䗍ⴓ倯玎䒭ך鑧

⡘縧鸞䏝⸇鸞䏝

⡘縧鸞䏝⸇鸞䏝ךꟼ⤘׾ֶׁ׵ְ׃גֶֻ

v (t) = dx(t) dt

a(t) = dv (t)

dt = d

²

x(t) dt

² x(t)

儗꟦tד䗍ⴓ

儗꟦tד䗍ⴓ

r(t)

v (t) = dr (t) dt

a(t) = dv (t)

dt = d

²

r (t) dt

² 儗꟦tד䗍ⴓ

儗꟦tד䗍ⴓ 如⯋麊⹛ך㜥さ 如⯋麊⹛ך㜥さ

䗍ⴓ倯玎䒭

d

²

x(t)

dt

²

= a t, dx

dt , x

v =

ֿ׸׾鍑ֻֿהד劢濼ךx tװv t׾実׭׷

⸇鸞䏝ך䞔㜠ָⴓַ׏גְ׷㜥さ

⸇鸞䏝ך䞔㜠

׍׳׏ה⯓《׶

չ麊⹛ך岀⵱ 麊⹛倯玎䒭պח״׷ה暟⡤ח⡲欽ׅ׷

⸂׾椚鍑ׅ׸ל暟⡤ח欰ׄ׷⸇鸞䏝ָⴓַ׷կ

⡘縧װ鸞䏝׾実׭׷ֿהָדֹ׷

侧㷕ה׃גך䗍ⴓ倯玎䒭

鍑כ㶷㖈ׅ׷ַ 㹀纏㚖כו׸ֻ׵ְ䎢ְַ

ⴱ劍勴⟝׾♷ִ׸ל鍑כ♧䠐ח㹀ת׷ַ

ⴱ劍⦼חꟼׅ׷鍑ך⣛㶷䚍כ

倯玎䒭ָػًٓ٦ة׾ろ׬㜥さך鍑ךػًٓ٦ة⣛㶷 䚍כ

׉׮׉׮鍑כⰧ⡤涸זꟼ侧ד邌ׇ׷ךַ

׃ַ׃ֿך䱇噟דכ麣Ⱗה׃ג䗍ⴓ倯玎䒭׾䪔׏גְֻ

䗍ⴓ倯玎䒭ךⴓ겲

杝甧㢌侧ך⦐侧⦐䌢䗍ⴓ倯玎䒭醱侧⨉䗍ⴓ倯玎䒭 倯玎䒭ָ劢濼ꟼ侧ֶ״ן׉ך㼪ꟼ侧חאְג如䒭ך 㜥さ׾简䕎䗍ⴓ倯玎䒭הְֲկ׉ֲדזְ׮ך׾ꬊ简䕎 䗍ⴓ倯玎䒭הְֲկ

倯玎䒭ָ劢濼ꟼ侧ֶ״ן׉ך㼪ꟼ侧׾ろ׬갪ך׫׾ろ

׬갪ך׫ַ׵ז׷׮ך׾俕如倯玎䒭׉ֲדזְ׮ך׾

ꬊ俕如倯玎䒭הְֲկ

葿ղזة؎ف׀הח鍑ֹ倯ךظؐعָ֮ؐ׷

简䕎הꬊ简䕎

♧菙חꬊ简䕎䗍ⴓ倯玎䒭כ鍑ֻךָꬊ䌢חꨇ׃ְ

鍑匿涸ז鍑ָ㶷㖈׃זְֿהָקה׿ו 简䕎ך⢽

ꬊ简䕎ך⢽

俕如הꬊ俕如

简䕎俕如倯玎䒭

简䕎ꬊ俕如倯玎䒭

暟⡤ך麊⹛ה䗍ⴓ倯玎䒭

琎ⴓ岀ה㢌侧ⴓꨄ

瘝⸇鸞䏝麊⹛ 䗁统

v(t) = adt = at + C

琎ⴓ㹀侧

d

²

x(t)

dt

²

= a

㹀侧

dv (t)

dt = a

֮׷ְכ

ֿ׸׵׾弫׋ׅ״ֲזx tװv t׾実׭׷ךָ湡涸

dv (t)

dt = a

ך⚕鴟׾tד琎ⴓׅ׷ה

琎ⴓ㹀侧׾ろ׬䕎ד剅ַ׸׋䗍ⴓ倯玎䒭׾弫׋ׅꟼ侧

׾׉ך䗍ⴓ倯玎䒭ך♧菙鍑הְֲկ 鍑

瘝⸇鸞䏝麊⹛ 䗁统

♧菙鍑׾実׭׋׌ֽדכ麊⹛ך圫㶨׾㸣Ⰻח✮鎉׃׋ה כְִזְկ麊⹛׾✮鎉ׅ׷׋׭חכ琎ⴓ㹀侧ך⦼׾寸 㹀ׅ׷䗳銲ָ֮׷կ

黝䔲זⴱ劍勴⟝ָ֮׸לv tָ㸣Ⰻח寸ת׷

⢽tךהֹחvv

v(0) = C = v₀ ״׶ v(t) = v₀ + at

v

t v

⫘ָֹa v(t) = at + C ♧菙鍑כ⫘ֹaך湫简ך꧊さ

搀侧ח֮׷

ד֮׷הׅ׷

ֿ׸׾弫׋ׅחכ

搀侧ך湫简ך⚥ַ׵ֿךאָ鼅ל׸׷

瘝⸇鸞䏝麊⹛ 䗁统

v(t) = v₀ + at

如ח⡘縧x t׾実׭׷

dx(t)

dt = v

₀

+ at

x(t) = (v

₀

+ at)dt = v

₀

t + a

2 t

²

+ C

ⱄן琎ⴓ㹀侧

黝䔲זⴱ劍勴⟝ָ֮׸ל琎ⴓ㹀侧ך⦼ָ寸ת׷

⢽tךהֹחxx

x(0) = v₀ 0 + a

2 0² + C = C = x₀

״׏ג

x(t) = x

₀

+ v

₀

t + a 2 t

²

♧菙鍑כ搀侧ך佝暟简ך꧊さ

ד֮׷הׅ׷

䗍ⴓ倯玎䒭ך鍑岀

⸇鸞䏝ָ儗꟦tךꟼ侧ה׃ג♷ִ׵׸גְ׷㜥さ ⸇鸞䏝ָxװvח⣛㶷׃זְ㜥さ

剑׮知⽃זة؎فך䗍ⴓ倯玎䒭

dv (t)

dt = a(t)

瘝⸇鸞䏝麊⹛כ

ֿ׸ך知⽃ז㜥さ 鍑׾実׭׷׋׭חכ⽃ח⚕鴟׾tד琎ⴓׅ׸ל״ְկ

v (t) = a(t)dt

x(t) = v(t)dt

ׁ׵חv׾tד琎ⴓׅ׸לxָ実ת׷

琎ⴓׅ׷嫣ח琎ⴓ㹀侧

ָ⳿גֻ׷

琎ⴓ㹀侧׾寸㹀ׅ׷חכאךⴱ劍勴⟝ָ䗳銲

⢽겗

⸇鸞䏝ָa(t)DPT tחז׷״ֲח麊⹛׃גְ׷暟⡤ָ֮׷կ

ֿך暟⡤כtךהֹחxNך⡘縧ד꫼姺׃גְ׋կ⟣䠐 ך儗ⵟחֶֽ׷ֿך暟⡤ך⡘縧ה鸞䏝׾実׭״կ

⢽겗

⸇鸞䏝ָa(t)DPT tחז׷״ֲח麊⹛׃גְ׷暟⡤ָ֮׷կ

ֿך暟⡤כtךהֹחxNך⡘縧ד꫼姺׃גְ׋կ⟣䠐 ך儗ⵟחֶֽ׷ֿך暟⡤ך⡘縧ה鸞䏝׾実׭״կ

dv (t)

dt = 2 cos(3t)

⚕鴟׾tד琎ⴓ׃ג

v (t) = 2 cos(3t)dt = 2

3 sin(3t) + C

tךהֹחvהז׷׋׭חכ

v (t) = 2

3 sin(3t)

״׏ג

v (0) = 2

3 sin(3 0) + C = C = 0

⢽겗

⸇鸞䏝ָa(t)DPT tחז׷״ֲח麊⹛׃גְ׷暟⡤ָ֮׷կ

ֿך暟⡤כtךהֹחxNך⡘縧ד꫼姺׃גְ׋կ⟣䠐 ך儗ⵟחֶֽ׷ֿך暟⡤ך⡘縧ה鸞䏝׾実׭״կ

v (t) = dx(t)

dt = 2

3 sin(3t)

⚕鴟׾tד琎ⴓ׃ג

x(t) = 2

3 sin(3t)dt = 2

9 cos(3t) + C

tךהֹחxNהז׷׋׭חכ

x(0) = 2

9 cos(3 0) + C = 2

9 + C = 1

x(t) = 2

9 cos(3t) + 11

״׏ג 9

鸞䏝ח嫰⢽׃ג幾鸞ׅ׷㜥さ

鸞䏝ח嫰⢽׃ג幾鸞ׅ׷㜥さ׾罋ִ׷

dv (t)

dt = kv (t)

姻ך㹀侧

ֿך㜥さכ⽃秪ח琎ⴓ׾׃ג鍑׾実׭׷ֿהָדֹזְկ ה׶ִ֮׆⚕鴟׾tד琎ⴓ׃ג׫׷կ

v (t) = ( kv (t))dt = k v (t)dt

劢濼ךꟼ侧 劢濼ךꟼ侧׾琎ⴓׅ׷ֿהזו♶〳腉

⡦ַⴽך倯岀ָ䗳銲

ꥡ简䕎俕如倯玎䒭

㢌侧ⴓꨄ

dv (t)

dt = f (v )g(t)

⚕鴟׾f vדⶴ׷

f (v 1 ) dv dt = g(t)

⚕鴟׾tד琎ⴓׅ׷

f (v 1 ) dv dt dt = g(t)dt

縧䳔琎ⴓׅ׷ה ^左辺 = 1

f (v) dv 1

f (v) dv = g(t)dt

ך䕎׾׃׋䗍ⴓ倯玎䒭חאְגכ㢌侧ⴓꨄהְֲ

ذؙصحָؙ⢪ִ׷կ

ֿ׸ז׵⚕鴟鎘皾דֹ׷

㢌侧ⴓꨄך穠卓ך鋙ִ倯

dv (t)

dt = f (v )g(t)

䊩鴟חvחꟼ⤘ׅ׷׮ך〸鴟חtחꟼ⤘ׅ׷׮ך׾꧊׭׷

dvהdt׮ל׵ל׵חׅ׷

1

f (v) dv = g(t)dt

⚕鴟ח琎ⴓ鎸〾׾ֻ׏אֽ׸ל⯓玎ךꟼ⤘䒭ָ䖤׵׸׷կ

1

f (v ) dv = g(t)dt

㼰׃׌ֽ湫錁涸ז椚鍑

dv

dt = lim

t 0

v t

鵚⡂ך礵䏝׾♳־׷הְֲ䠐㔳

þtָ㼭ֽׁ׸ל鵚⡂涸ח

v t f (v )g(t) v

f (v ) = g(t) t

ֿ׸ָþtװþvך稢ַזⴓⶴ׀הח䧭׶甧׏גְ׷ַ׵

v

f (v) = g (t) t

鵚⡂ך礵䏝׾搀ꣲח״ֻׅ׷

lim

v 0

v

f (v ) = lim

t 0

g(t) t

1

f (v) dv = g(t)dt

⚕鴟ך駈׃♳־

㢌侧ⴓꨄ

ꥡ简䕎俕如倯玎䒭ך㜥さחֿך䩛岀׾黝欽ׅ׷

ꥡ简䕎俕如倯玎䒭 ך鍑ךⰕ䒭

׋׌׃K tכĜ tך⾱㨣ꟼ侧 琎ⴓ㹀侧

MPHvָ実ת׏׋կ⚕鴟׾Fך肔חךׇ׷ה

Ae^Cה琎ⴓ㹀侧׾縧ֹזֶ׃׋

鸞䏝ח嫰⢽׃ג幾鸞ׅ׷麊⹛

dv (t)

dt = kv (t) 1

v dv = kdt

⚕鴟׾琎ⴓ׃ג

1

v dv = k dt

log v = kt + C

⚕鴟ַ׵⳿גֻ׷琎ⴓ㹀侧׾

〸鴟ח꧊׭׋

v = e ^kt+C = e^Ce ^kt

⚕鴟׾eך肔חךׇ׷

鸞䏝ח嫰⢽׃ג幾鸞ׅ׷麊⹛ך㜥さ

v = e ^kt+C = e^Ce ^kt ⴱ劍勴⟝ך⢽tךהֹחvv

״׏ג v(t) = v₀e ^kt v(0) = e^Ce ^{k 0} = e^C = v₀

t v

v⁰

➙䏝כ⽃秪ח琎ⴓׅ׸לxָ実ת׷կ

鸞䏝ח嫰⢽׃ג幾鸞ׅ׷麊⹛

x(t) = v (t)dt = v

₀

e

^kt

dt = v

₀

k e

^kt

+ C

ⴱ劍勴⟝ך⢽tךהֹחxx

x(0) = v

₀

k e

^{k 0}

+ C = v

₀

k + C = x

₀

x(t) = v

₀

k 1 e

^kt

+ x

₀

״׏ג

t x

x₀ + v₀ k

x₀

鸞䏝ח嫰⢽׃ג幾鸞ׅ׷麊⹛

㉏겗

如⯋ך麊⹛׾罋ִ׷կ儗ⵟtחְֶגxחְג

vv ד麊⹛׃גְ׷暟⡤ָ֮׷կֿך暟⡤ָ鸞䏝ך

⛦ח嫰⢽׃ג幾鸞׃גְ׷㜥さח⟣䠐ך儗ⵟtחֶֽ׷鸞 䏝ה⡘縧׾実׭״կ׋׌׃⸇鸞䏝ך鸞䏝ך⛦ח㼎ׅ׷

嫰⢽⤘侧ך㣐ֹׁ׾k הׅ׷կ

ת׋ֿך麊⹛ך圫㶨׾鸞䏝ח嫰⢽׃ג幾鸞ׅ׷㜥さה 嫰鯰ׇ״կ