24.15章.微分方程式

m d y dt = F m d y = mg dt V y = dy dt d y dt = d dy dt dt = dv y dt

dv y dt = g dv y dt = g dt dt dv y = g dt V y ( t) = gt + C V y ( ) = V y ( ) = C = V y t ( ) = gt V y ( t) = dy dt = gt dy = g t dt y( t) = g t + C y( ) = H y( t) = g t + H t y= = H / g V y t y= ( ) = gt y= = gh

θ x axis: m d x dt = y axis: m d y dt = mg V x = dx dt = C θ C = V o cos V x = dx dt = V cos o θ x = ( V o cos )t V y = dy dt = gt + C θ V y = dy dt = gt + V sin o y = g t + ( V o sin )t t = x V o cos

y = g x V o cos ( x + V o sin V o cos ( = g V o cos ) + x ( V o cos ) ( ) * g ( x V o cos ), ( ), g g y = V o cos g = V o cos * + x V o ( ) g = V o cos x V o ( ) cos sin ( g ) sin ( g ) sin ( cos ) V o cos g ( + V cos o ) g + V o sin g,+ ( ) sin 3 cos 4 5 sin - ( / cos / ). sin cos x -+. /+ θ Y[m] () V o () () X[m] θθ θ)=θ)

() *+,- (, +)9 ()* +,-./ 3456748 3456748: θ m d ( L ) = mgsin dt sin = 3 3 + 5 5 7 7 sin + L d = g dt o = g / L d dt + = θ = o cos( o t) d dt = o o sin( o t) d dt = o o cos( o t) = o o = g L f = o = g L T = f = L g sin = 3 / 3

dx dt = kx dx x = kdt log e x = k dt = kt + C x = e kt+c = e C e kt x = x o = e C x = x o e kt dn dt = kn dn / N = kdt log e N = kt + c N t ( ) = N o e kt N / N o = / e kt.5 = / k = log e / t.5 ( ) N t ( ) = N o e log e t /t.5 t = { log e ( N o / N ) / log e }t.5 t.5 K(/s) / / / 3 8 x -6 6 53 8 34.x -8 6.6 3.3 37 3 7.3x - 3 9 3 7.3x - 3

k = log e / t.5 =.693/ / t.5 k =.693/ / 8 [ / days] =.8665 [ / days] t = log e ( N o / N ) = k log e ( ).8665 [ / days] = 4.65 = 53.6 [days].8665 [ / days] dn dt = kn dn dt = { k( N )}N dn N N ( ) = kdt N + ( dn = ( N ) ( kdt log N log N N N = ekt +C N = e kt +C Ne kt +C e kt +C N = kt +C e ( ) ( ) = kt + C

N o N o = e C N = ec e kt = N o e kt e C e kt N o N o e kt N o N = o e kt ( N o N o e kt ) L di + RI = V dt I = = L / R

di dt = V L I = V L I ( di I V L ( log I V L = dt ( = t + C I t ( ) = V L + ec e t e C = I o ( ) = V L + I oe t I t I ( ) = V / L + I o = I o = V / L ( ) = V L e I t t ( = V R e t ( = L / R I = V / R = L / R =.5 s I=V/R I=V/R(- Exp[-(t-5)/τ I=V/R Exp[-(t-5)/τ I = V / R ( ) = V L + I o e I t t I = I o e t [A] = [A][]

I o = V / R I = dq / dt RI + Idt C o = V I = dq / dt R dq dt + Q C o = V dq dt ( ) = Q C ov RC o dq ( Q C o V ) = dt RC o log( Q C o V ) = t + c Q( t) = e c e RC o t RC o + C o V Q( ) = = e c + C o V Q( t) = C o Ve t RC + C o V = C o V e t RC o I ( t) = dq dt = C o V e RC o t RC o = V t R e RC o

L di dt + C o L d I dt o = / Idt = V + I C o = LC o d I dt + o I = RI = V o sin t ( ) () * I = V o sin (t) R Q C o = C o Idt = V o sin (t) I = V C o cos (t) o

( ) I = C o V o cos t ω L di dt = V sin t o ( ) I = V o L sin (t) dt = V o L cos (t) ω L di dt + RI = V o sin t ( ) e it = cos (t) + isin (t) ω V = V o e it I = I o e it

L d ( I o ) eit + R( I o e it ) = V o e it dt LI o ie it + R( I o e it ) = V o e it V o = I o ( R + i L) ()*+ V = V o e it = I o ( R + i L)e it I = I o e it tan = L / R δ L di dt + RI = V o sin (t) I = I o sin (t) V o sin (t) = LI o cos (t) + RI o sin (t) = I o { Rsin (t) + L cos (t)} = I o R + ( L ) sin (t + ) cos = R ( ) sin = L R + ( L ) R + L tan = L / R δ V o = I o R + L ( ) C Idt + RI = V o sin (t) V = V o e it I = I o e it

C I o eit dt + R( I o e it ) = V o e it I o ic eit + R( I o e it ) = V o e it *+,-.) ( V o = I o R + ic V = V o e it = I o ( R i / C)e it I = I o e it tan = /CR δ C Idt + RI = V o sin (t) I = I o sin (t) () * V o sin (t) = C I o cos (t) + RI o sin (t) = I o Rsin t ( ) cos ( t ) C ( = I o R + sin t + ) C ( ) ( ) R cos = R + (C) sin = C R + C ( ) tan = /CR δ ( ) V o = I o R +/ C di dt = V o cos t ( ) I C ( / R

α alpha β beta γ Γ gamma δ Δ delta ε epsilon ζ η eta θ Θ, theta ι iota κ kappa λ Λ lamda µ mu ν nu ξ Ξ ( xi (ksi) ο omicron π Π pi ρ rho σ Σ sigma τ tau υ Υ upsilon φ ϕ Φ phi χ chi ψ Ψ psi ω Ω omega

π

σv π

34 = 5 3 + 4 5 3 + 4 5 + 5 + = 444 5 34 = 3 6 + 3 5 + 5 3 + 3 3 + 3 + 3 + = 3 6 3 8.53 /.5 5 5 3 5 / 8 / / 4 3 8 4 7 5.565 3 / 4 / 3 / 5 / / 5 7 / 8 4 / 7 a = kb c = kd kb + b k + kd + d = = kb b k kd d a / b + c / d + = a / b c / d ( ) = x + 4x + 4 ( x 3) = x 6x + 9 ( x + 4) ( x 4) = x 6 ( )( x + 6) = x +x + 3 ( x +) ( x + ) ( x + 3) = 6 + x + 6x + x 3 ( ) x + 5x + 6 = ( x + ) ( x + 3) x 5 = x 5 ( )( x + x +) x + x + 5 x 4x + 4 = x x + 5 = x + i 5 ( )( x i 5) x3 + 9x + 9x + 8 = x + 8 y = / 3x S = t + 5 ( )( x + 5) [km / h]v = 36 [km / h] ( t =.485[h] y = 7tV = 8.[km / h]

Cs = 34 [m/s] L = C s t = 34[m / s] 3 = [m] ( =.[m] =.375[m] V = R = 637[km] / 4[h] = 667 [km/h]=463[m/s]=.36 x = / 3 x = 4 x = 3 t = t = 6 x > 4 t < t < 6 x = 5 /7, y = 4 /7 x = / 4, y =, z = / 4 y = x y = x + 5 y = x ( ) + 5 y = x x + x = ( / )t = t = / 4 = 7.7 s V = t = 4 7.7 = 8.8 [m / s] 5 ( ) / () ( 7 5) / (3) ( + 4 7 ) / 3 i i i x = 3 ± i 5 x = ± i x = 3 ± 5 x = ± ( ) / 8 x = 3 ± i 7 ( ) / x = ± 5 3 + i 3 i 5 + i / 5 6 4 4 4 ) 7 64 +4 6 8 64 + 64 9 8 9 ) 96 4 + 9 8 8 8 5 4 + 8 5 4 5 = 5 + = 4 + 5 ( ) 5 = + 4 + 5 ( ) = + 4 + 4 + 4 + 4 +...

6 = 6 + = 4 + 6 ( ) 6= + log x log a x m( x) = m 3 log( x) = log 3 =.477 x =.477/ =.9748=.58 = r =.5 /8 (.)..8 8 4 + ( 6 ) = + + 4 + + 4 + + 36.5 365 5 + = = = ( ) 3 =.57[m] 5 /8(radian) 7 /8(radian) = r = 3.459 3.4[m] S = r =.785398.7854[m ] + = 5 + 3 = 4 = + = ( ) = sin ( / 4) = / sin ( ) = cos ( / ) = ( ) = cos( 3 / 4) = / tan ( ) = tan ( / 4) = ( ) = tan( 7 / 4) = sin / cos tan 3 / 4 sin ( ) = 5 ( ) / 6 cos ( + ) = ( 3 5 ) / 6 ( ) = ( 4 5 3) / 7 tan = / 3, tan = 5 / tan +

sin 4 cos = { sin( 4 + ) + sin ( 4 )} = { sin( 6 ) + sin ( )} cos sin = { sin( + ) sin ( )} = { sin ( 3 ) sin ( )} cos cos 4 = { cos( + 4 ) + cos ( 4 )} = { cos( 6 ) + cos ( )} sin sin = { cos( + ) cos ( )} = { cos ( 3 ) cos ( )} cos = a / a + b = / sin = b / a + b = 3 / tan = 3 = / 3 sin + cos = sin + / 3 ( ) a sin A = b sin B = c sinc = R 6 cm sin 3 A = + + 3 8 = 3o B = a sin 3 o = b sin6 o = + + 3 8 = 6o C = 6 cm = R R = = 6 cm o o sin 3 3 + + 3 8 = 9o c sin9 o a :b : c = : 3 : = : 3 : a sin A = b = R sin B m sin o = b sin 3o b = o sin 3 sin = o = 3 3 3 m S = absinc = 3 3 sin 3o = 5 3 = 8.86 m 3 ) ( s = a + b + c = 5 + 6 + 7 = 9 m

S = { s a} { s b} { s c}s = { 9 5} { 9 6} { 9 7}9 = 4i3ii9 = 4.69 m sin = 6, sin = 3 4, cos = 6, cos = 4 3, tan = 6 tan 3 = 3 ~ /

)* ( U + V = + 5 =.8 km/h A x = cos(3 o ) = 5 3A y = sin(3 o ) = 5 cos = A x B x + A y B y + A z B z A x + A y + A z B x + B y + B z = A x + A z A z B z B y + B z = 4 + 4 4 + 4 = = 6 o V = A x B x C x A y B y C y A z B z C z = = 8 + 4 + 4 = 6m 3

A B = ( A B A B y z z y )i + A z B x A x B z = 4i 4j + 4k ( )k ( ) j + A x B y A y B x A B = 4 + 4 + 4 = 3 4 = 4 3 E = ( A B) E ( C D) = ( EiD)C ( EiC)D t = aib bib = a xb x + a y b y + a z b z 3 + = b x + b y + b z + +.5 =.5 c = a + tb ( ) =. ( ) =.88 ( ).5 =.44 c x = a x + tb x = 3 /.5 c y = a y + tb y = /.5 c z = a z + tb z = + /.5 L = c x + c y + c z = 3.34 +.5 + x +.5 +.5 + y + +.5 + z = 7.5 +.5 + = 3.6 a = 3.6 R I + R I I R I I ( ) = V V ( ) + R 3 I = V I = 4 = 8 + 4 = 9 3 = 3 I = 4 = + 4 4 = 6 3 = 4 3 8 8 9 4 5 () line () line (3) line 4 () line 5 (3) line 3 () line = 5 = () - (3)

S = / 3 S = / 9 S = 9 S = B / ( B A) S = 9 / 8 S = / S = 9 /9 S = A / (A + B) dy / dx =.6 dy / dx = 4 / 5 dy / dx = x dy / dx = x 3 dy / dx = 5x dy / dx = ( 3 / ) x 6x + a 6 + 6a 5 b + 5a 4 b + a 3 b 3 + 5a b 4 + 6ab 5 + b 6 dy / dx = ax + b dy / dx = 3ax + bx + c dy / dx = / x dy dx = a x + 4b 3 x dy dx = anxn bcx c 5 dy y = 7x 5 + 6x 3 + 9x y = ( 7x 5 + 6x 3 + 9x + ) / ( 3x + ) y = 8( 3x + x + ) 3 ( 3x + ) dx = dy mxm dx = 6x5 5x 4 + 9x x y = x / 4 x y = 6x / ( x + ) 4 y = x + 4 + x / x + 4 dx dy = 3y dy dx = 3y = 3x 3 dx dy = 4y3 dy dx = 4y 3 = dy / dx = sin x dy / dx = 6cos 3x dy / dx = 6 / cos 3x y = 5sin 4 x cos x y = 3cos 3 x + y = 6xsin( 3x +) sin y = x cos y dy dx = x dy dx = x cos y = x sin y = x x cos y = dy sin y x dx = x 3 dy dx = x 3 cos y = x 3 x / a ( ) = x 3 a tan y = sin x a x 4x 3 4

( + tan y) dy dy = cos x dx dx = cos x + sin x dy / dx = x log dy / dx = 5 x log 5 u = sin x y = e dy u dx = dy du du dx = eu cos x = e sin x cos x u = x y = e dy u dx = dy du du dx = eu x = xe x log y = ( log x) / x d log y dx = y dy dx = x log x + x x = x ( log x) dy dx = x x x ( log x) log y = sin x y log y = x x y dy dx = dx x dx = x x dy dx = dy x dx = ( + log x) y x = e sin x x dy / dx = yx x ( + log x) = e xx x x ( + log x) y = 4x L = x + y ( ) S = xy = x( L / x) = L / x x ds / dx = L / x = x = L / 4 V = RI = ri = R p I R p = Rr / R + r I = V / r = I R / R + r ( ) P = ri = r I R / ( R + r) ( ), ( ( )), ( ( )) dy / dx = sin x = cos x + / d y / dx = cos x = cos x + / y (n) = cos x = cos x + n / b y () ydy = a b kdx = k dx = k x a b a b a = b ( a ( ) = R I r / R + r y = sin x = cos x + 3 / [ ] b a = k( b a) df (x) 5 t () tdt = 5 ( ) ( ) ( ( )) b = f (x) a = 5 ( = 5 [ ] a b = f (b) f (a)

a x dx = a x 3 3 ( +a a y = y = x dx = y = = a3 a z 3 3 z dz = 3 ( a m + xm+ + C y = t + +a a = a3 5 3 t t 3 dt = 3 3 3 x x + C y = x dx = 5 5 x + C x dx = x + C y = x dx = x + C y = 7 x7 5 x6 + x 3 x + C y = 5 x5 + 3 3 x 3 + x + C y = x + x x 5 = 5 3 ( x) + x = + 3 3 4 3 4 5 = + x + x + x + x + x 3 4 5 3 4 5 3 4 5 6 6 7 + x + x + ii 6 7 3 5 4 7 5 6 33 7 = + x x + x x + x x + x 8 6 8 56 4 48 ( ) 3 3 4 5 5 7 6 7 33 8 y = + xdx = + x = x + x x + x x + x x + x + 3 8 3 6 4 8 5 56 6 4 7 48 8 x y = cos( x) dx = x sin dx = ( ) + C 3sin ( x ) sin x dx = 3 cos x dx = x sin x + C cos ( x ) + C xsin xdx = x( cos x) dx = x cos x cos x = x cos x + cos x + C x sin xdx = x ( cos x) dx = x cos x ( x)cos xdx = x cos x + x( sin x) dx = x cos x + xsin x sin xdx = ( x )sin x + xsin x + C t = x + a y = y = t dt = 4 ( ) 3 x + a 3 + C t = ax + b dt = adx sint dt cos(ax + b) t = + C t = x + a dt = xdx y = a a dt = 4 t + C

t = ax + b dt = adx y = dx = x x 3 x dx = x 3 3 3 t a dt = a t + C ( x 3) ( x +) dx = 4 x + x 3 ( dx = log x + log x 3 4 = 4 log x 3 x + + C ( ) x 3 3 d x 3 3 = 3 log x3 3 + C ( ) cos x dx = d + sin x + sin x = log + sin x + C + sin x x e x dx = x ( e x ) dx = x e x xe x dx = x e x xe x e x dx = ( x x + )e x + C ( ) x 3 e x dx = x 3 e x dx = x 3 e x 3 x e x dx = x 3 e x 3 x x + ( ) e x + C x ( x ) ( x +) dx = x ( x ) x ( ( x +) ( dx = ( x ) dx + ( x ) dx = log x x ( ) log x + ( + C x ( x +) dx ( ( x 3 dx = x x ( x +) dx = x x + ( +) x + ( ) ( ) + x x + dx = ( ) x ( ) + log x + + C x + + x = t x + x = t + t x = t t dx = t + dt t x + x dx = t t t + dt = t t + t t dt = t t + ( dt = log t log t + = log t t + tan x / ( ) = t dx = + sin x + cos x dt + t + t + t + t = + t dt = + t + t + t dt = log + tan x + t + C

S = L = / / d asin cos 4 = 4 d = a d = a ( sin 4 4 ) * r dx / dt = 3a cos t sint dy / dt = 3asin t cost / S = ( ) ( 3acos t sint ) + ( 3asin t cost) dt = 3a cos t sin t ( cos t + sin t) dt / = 4 3a sint costdt = a sint d sint S = ( x x) dx = 3 3 x. r / x ( / ( ) = a sin t = 3 = 6 d asin cos 4 = 4 d = a d / = a ( sin 4 4 ) * ( ) / = a f x = f / x = y f y = f / y = x / ) ( / = 6a / = a c c S = x + y + dx dy = d r + r dr = u du = 3 3 u ) ( c + = 3 ( c + ) 3 * ( ) x = r cos, y = r sin u = r + du = rdr S = / ( )( Rd ) Rsin = R sin d = 4 R cos / = R /3 y = / /3 [ ] /3 R x x G = R y xdx dy S = xydx R / = 4 R R 4 x R x dx R = R x d(r x ) R ( ) y = * R, 3 R x + ( ) 3 R x = t - /. R = R 3 *+ R3 -. = 4R 3 =.48R M = SL S = M / L I = m i r i = x dxs = M L i L/ L/ ( ) L/ L/ x dx = M L x 3 3 ( * ) L/ L/ = M ( L L / ) 3 = ML 3

r x = x x + y + z = r = z y y ( z ) * i + x z z ( x ) * j + y x x ( y ) * k = r = x x + y y + z z = 3 z = e i x x + y + z = x r r y = y r r z = z xi + yj + zk r = = r r r r e i ( + ) = cos + ( ) + isin ( + ) { }{ cos + isin } = ( cos cos sin sin ) + i( sin cos + cos sin ) = e i e i = cos + isin V y = gt +V o = t = V o / g y = ( g / )t +V o t = t = V o / g V y = g V o / g RdQ / dt + Q / C o = Q t ( ) = e c e t RC o Q( ) = C o V C o V = e c Q( t) = C o Ve L di dt + RI + C Idt = V o sin t ( ) -->V o = i L + R + ( ) +V o = V o t RC o ic o I = R + i L ( o C o I o L C o ( ) I o