x ( ) x dx = ax

x ( ) x dx = ax

1 dx = a x log x = at + c x(t) = e at C (C = e c ) a > 0 t a < 0 t 0 (at + b )

h dx = lim x(t + h) x(t) h 0 h x(t + h) x(t) h x(t) t x(t + h) x(t) ax(t) h x(t + h) x(t) + ahx(t) 0, h, 2h, 3h,...

Mathematica a=1;h=0.1; For[i = 1; x = 1; kinji = {{0, x}};, i <= 10, i++, x += a*x*h; AppendTo[kinji, {i*h, x}]]; ListPlot[kinji,PlotJoined >True]

dx = x Figure: 1 1 2 1 0.2 0.4 0.6 0.8 1-1 -2

dx = ax bx2

h ( ) Figure: h = 0.1 ( ) 2.75 2.5 2.25 2 1.75 1.5 1.25 0.2 0.4 0.6 0.8 1

2 2 dx dy = ax + by = cx + dy

dx dy = ax + by = cx + dy d ( ) a b A = c d ( ) x = A y ( ) x y

( ) ( ) ( ) λ 0 X x A P 1 AP = = P 0 µ Y 1 y ( ) d X = d ( ) ( ) ( ) x Y P 1 = P 1 x A = P 1 X AP y y Y dx dy = λx = µy 2

dx dy = 7 3 x 2 3 y = 1 3 x + 8 3 y A = ( 7 3 2 3 1 3 8 3 )

A = {{7/3, 2/3}, { 1/3, 8/3}} Eigenvalues[A] Q=Eigenvectors[A] Q P=Transpose[Q]

Inverse[P].A.P ( 3 ) 0 0 2 ( ) ( ) X = P 1 x Y y dx dy = 3X = 2Y

dx dy = x y = x + y

Mathematica h = 0.1; For[i = 1; x = 0.1; y = 0; kinji = {{x, y}}, i <= 67, i++, x += h(x - y); y += h(x + y); AppendTo[kinji, {x, y}]]; ListPlot[kinji, PlotJoined > True, Ticks > None]

x y x += h(x -y); y += h(x + y); x y x

Mathematica x x u y x u x h = 0.1; For[i = 1; x = 0.1; y = 0; kinji = {{x, y}}, i <= 67, i++, u=x+ h(x - y); y += h(x + y);x=u; AppendTo[kinji, {x, y}]]; ListPlot[kinji, PlotJoined > True, Ticks > None]

Figure: (0.1, 0)

A = ( ) 1 2 1 1 A dx dy = x + 2y = x + y

F = ma 3 F = G mm r 2

mg m d 2 y 2 = mg dy dv = v = g v y 0 v y 0 v(t) = v y 0 gt y(t) = y 0 + v 0 t 1 2 gt2

x d 2 x 2 = 0 x x 0 v x 0 x(t) = x 0 + v x 0 t t y(t) = y 0 + v y 0 v0 x (x(t) x 0 ) 1 ( ) x(t) 2 2 g x0 v0 x

Mathematica x, y 4 x, y g=1;h = 0.1; For[i = 1; x = 0; y = 0;vx=0.5;vy=20; kinji = {{x, y}}, i <= 100, i++,ux=x+ h*vx; vx += 0; uy=y+h*vy;vy+=-g*y; x=ux;y=uy; AppendTo[kinji, {x, y}]]; ListPlot[kinji, PlotJoined > True, Ticks > None]

( ) m dx 2 2 = kx dx = v dv = kx v x(t) = A sin(bt + C) x(t) = A sin( kt + C) v(t) = A cos( kt + C) A, C

m d 2 x = mg sin x 2 x 0 sin x x

dx dy dz = 10x + 10y = 29x y xz = 8 3 z + xy xy

Mathematica h = 0.01; For[i = 1; x = 0.1; y = 0.1; z = 0.1; kinji = {{x, y}}, i < 5000, i++, u=x + h(-10x + 10y); v=y + h(28x - y - x*z); z += h(-8z/3 + x*y); x=u; y=v; AppendTo[kinji, {x, y}] ] ListPlot[kinji, PlotJoined > True]

20 10-15 -10-5 5 10 15 20-10 -20

dx dy = y = εy(1 x 2 ) x

h = 0.1; m = 2; For[i = 1; x = 0.1; y = 0; kinji = {{x, y}}, i <=1000, i++, u =x+ h*y; y += h(m*(1 -x 2 )*y - x); x=u AppendTo[kinji, {x, y}]]; ListPlot[kinji, PlotJoined > True, Ticks > None]

Figure:

x(t + h) x(t) + x (t)h x(t + h) x(t + h) = x(t) + x (t)h + x (t) 2 h2 + 2

y = f (x, y) h y(0) x T, 0, h, 2h,... T /h m y m x m = mh l y m y m y m l+1 y 0 = y(0) y 1,..., y l 1 y m+1 y m+1 = f (x m+1, y m+1 ) y m+1

( ) ( ) y m+1 = y m + hy m (y m = f (x m, y m )) y m+1 = y m + h 2 (y m + y m+1) y m y m y m+1 h h

x m < x < x m+1 y(x) = y m + (x x m )y m y m+1 = y m + h 2 y m + h 2 y m+1 = y m + h 2 y m + h 2 (y m + hy m + ) = y m + hy m + h2 2 y m + y(x) = y m + (x x m )y m + (x x m) y m + 2 2 y m+1 = f (x m+1, y m+1 ) y m+1

y m+1 = y m + hy m+1 y m+1 = y m 1 + 2hy m y m+1 = y m hy m + h2 2 y m + + 2hy m = y m + hy m + h2 2 y m + 2

y 1 y 1 z y 1 = y 0 hy 0 z = y hy y w w = y y y = z + 2h y y y 1 z = w

2 y (x) y (x+h) y (x) h y (x) y (x h) h h = y (x + h) 2y (x) + y (x h) h 2 2 2 2