(Differential Equations) - ncue.edu.tw

1 (Differential equations ) 1 ( 19 ) ( ordinary differential equation, ODE) x y=f(x) y(n)(x) =f(n)(x),n N :F(x,y(x),y (x),..,y(n)(x)) = 0 (1) , y,y ,..,y(n) , F , (1) (linear) ; F y,y ,..,y(n) , (1) (nonlinear) (1) y,y ,..,y(n) n (order) F(x,y,y ) = 0, F(x,y,y ,y ) = 0 , (1) , : , x(t) y(t) , , (system) u(x,y) u x, u y , (partial differentialequation, PDE) , , , , , , , 1 1.

2 :(a)y + 3y= 0(b)y + 3yy = 3x(c)y(4)+ 3y= ex+ sinx(d) x (t) = 3x(t) + 2y(t)y (t) = 2x(t) y(t)(e) u 2u x2+ 2u y2= 0 y=f(x) (1), y=f(x) (solution) ( y=f(x) ) , , ( , ) , , , , , , , , , , , , 2 (Separable equations , 42 ) dydx=p(x)q(y),(2) , (2) (separable equations ) , y (x), p(x) q(y) : q(y)6= 0 , (2) 1q(y)dydx=p(x) Z1q(y)dydxdx=Zp(x) dx Z1q(y)dy=Zp(x) dx,2 y=y(x) dy=y (x) dx=dydxdx , y , x , Q(y) 1q(y) ,P(x) p(x).

3 Q(y) =P(x) +C Q(y) P(x) =C, C , C , C, (2), , (initial value problem), y(x0) =y0 , , p(x) q(y) , x0 , C , , (differential form) ; (2) , :1q(y)dy=p(x) dx p(x) dx+ q(y) dy= 0, q(y) = 1q(y) 1( 43 ). dydx=x21 y2 . 2. yy =xy2+y2+x+ 1,y(0) = 1 . (Homogeneous equations , 49 ) (homogeneous equation):dydx=F yx :(a)dydx=y2+xyx2= yx 2+yx (b)dydx= lnx lny+x+yx y= ln yx +1+yx1 yx v(x) =y(x)x, v(x) y(x) =v(x)x, dydx=dvdxx+v F(v) =dvdxx+v dvdx=1x(F(v) v) =p(x)q(v), p(x) =1x,q(v) =F(v) v.

4 (homogeneous) , : , 1( 50 ). dydx=y 4xx y . s t {Cs} { Ct}, {Cs} { Ct} , , , (orthogonal trajectory) , , 4 2. {Cr:x2+y2=r2,r >0} { Cm:y=mx,m ( , ]} , Cm= x= 0 . 3. {Ca:x2+y2=ax,a >0} (Linear equations , 31 ) dydx+P(x)y(x) =Q(x)(3) , (3) (linear equation) : , , , , I(x), (integrating factor), I(x)dydx+I(x)P(x)y(x) =I(x)Q(x), I(x)P(x) =I (x), I(x)dydx+dIdxy(x) =I(x)Q(x) ddx(I(x)y(x)) =I(x)Q(x), x , I(x)y(x) =ZI(x)Q(x) dx+C y(x) =RI(x)Q(x) dx+CI(x), C (3) , I(x)P(x) =I (x), : I(x)?)

5 , :dI(x)dx=I(x)P(x) 1I(x)dI(x)dx=P(x) Z1I(x)dI(x)dxdx=ZP(x) dx Z1I(x)dI(x) =ZP(x) dx ln|I(x)|=ZP(x) dx |I(x)|= e P(x) dx I(x) , , , I(x) = e P(x) dx , (3) dydx 1 , dydx 1 , , : I(x) = e P(x) dx 6 1( 33 ). y +12y=12ex3 . 2( 77 ). (Bernoulli s equation) :dydx+p(x)y=q(x)yn, n R n= 0, dydx+p(x)y=q(x) n= 1, dydx+p(x)y=q(x)y, dydx+ (p(x) q(x))y= 0, n6= 0,1, v(x) =y1 n(x), v(x) (a) n6= 0,1 (b) y 2y= e 3xy 2,y(0) = 2.

6 7 , dydx 1 , (3) , n(x)dydx+p(x)y(x) =q(x) (4) (3) , P(x) =p(x)n(x) Q(x) =q(x)n(x) (4) i(x) i(x)n(x)dydx+i(x)p(x)y(x) =i(x)q(x), (i(x)n(x)) =i(x)p(x), (i(x)n(x)y(x)) =i(x)q(x), i(x)n(x)y(x) =Zi(x)q(x) dx+C y(x) =Ri(x)q(x) dx+Ci(x)n(x) i(x):(i(x)n(x)) =i(x)p(x) i (x)n(x) +i(x)n (x) =i(x)p(x) i (x)n(x) =i(x)(p(x) n (x)) ddxln(i(x)) =p(x) n (x)n(x), i(x) = e p(x) n (x)n(x)dx dydx 1 I(x) Zp(x) n (x)n(x)dx=Zp(x)n(x)dx Zn (x)n(x)dx=Zp(x)n(x)dx lnn(x), i(x) = e p(x) n (x)n(x)dx= e p(x)n(x)dx lnn(x)=e p(x)n(x)dxn(x)=e P(x) dxn(x)=I(x)n(x), y(x) =Ri(x)q(x) dx+Ci(x)n(x)=RI(x)n(x)q(x) dx+CI(x)=RI(x)Q(x) dx+CI(x) , , , dydx 1 I(x) 8 3.

7 : (cos2xsinx)y + (cos3x)y= 1,y( 3) = 1 . (Mathematical Models, 51 ) ? ? , , , , , : , , , ( ) , , , , , (Population Growth, 79 )1798 (Malthus) : , dPdt=kP,(5) P(t) t , k (5), ? ? , !

8 9 1. :P (t) =kP,P(0) =P0 . 2. 20 ( : ) 1951 1961 1971 1981 1991 2001 361 439 548 683 846 1029(a) 1961 and 1981 2001 , (b) (a) 2020 . 1961 t= 0, 1981 t= 20 P(t) =P(0)ekt= 439 ekt, P(20) = 439 e20k= 683 k= , P(t) = 439 2001 2020 P(40) = 439 40= 439 1063P(59) = 439 59= 439 1617 . (5) ? (Radioactive Decay) : , m0 t0 , m(t) t0 t , , :dmdt=km, m(t0) =m0, k (half-life) , , k 3.

9 14 5730 , 14 74%, . (Newton s Law of Cooling) (Newton s Law of Cooling) , T(t) t , Ts ( ), T(t) :dTdt=k(T Ts), k , :d(T Ts)dt=k(T Ts), T(t) :T(t) Ts= (T(0) Ts)ekt T(t) =Ts+ (T(0) Ts)ekt T(0) , ; T(0) , , 11 4( ). , 1:30PM C, , C C, C .. , , , ? ( < < ) (Continuously Compounded Interest, 55 ) A0, r, , t (n ):(a) : = (1 + ),A=A0(1 +rt) (b) : = 1 + ,A=A0 1 +rn nt (continuously compounded interest) n , A(t) = limn A0 1 +rn nt= limnr A0 1 +1nr nr rt=A0 limy 1 +1y y rt=A0ert :dAdt=rA 5.

10 6% , ? . (Logistic Models, 80 ) , , , , : , ; , ? (lo-gistic model) ( Verhurst) 1845 :dPdt=kP 1 PM ,(6) M >0 (carrying capacity), 6. (6) . 7. (6), P(0) =P0 (0,M) . , , , ( ), : , , , , , : , , , 8.