工程數學問題~~ 請給我詳細的解說謝謝

1.自由落體向下=+軸向,重力加速度=g,導出t秒時落體的位置函數為: y(t)=0.5gt^2

g=dv/dt => v=gt+c1v(0)=0+c1=0 => c1=0v(t)=gt=dy/dt=> y(t)=gt^2/2+c2y(0)=0+c2=0 => y(t)=gt^2/2 2.用分離變數法解ODE:y'=1+y^2 => dx=dy/(1+y^2)x+c=∫dy/(1+y^2)=atan(y)
3.ODE: -ydx+xdy=0 , 求積分因子F(x)並解此ODE.
Hint:R(X)=1/Q(Py-Qx) R'(Y)=1/P(Qx-Py) 0=(xdy-ydx)/x^2=d(y/x)c=y/x=> y=c*x & F=1/x^2 4.解線性ODE之初值問題: y'+y*tan(x)=sin(2x); y(0)=1F=e^∫tan(x)dx=e^∫sin(x)dx/cos(x)=e^[-∫d(cos(x))/cos(x)]=e^[-ln(cos(x))]=1/cos(x)F乘入原式裡面:{y/cos(x)}'=sin(2x)/cos(x)y/cos(x)=0.5∫sin(x)cos(x)dx/cos(x)+c=0.5∫sin(x)dx+c=-cos(x)/2+cy=-{[cos(x)]^2}/2+c*cos(x) 5.ODE為xy"+2y'+xy=0 已知其中一解是 y1=cos(x)/x ,求另一解 y2y1=cos(x)/x => x*y=cos(x)微分1次: xy'+y=-sin(x)微分2次: xy"+2y'=-cos(x)=-x*y => xy"+2y'+xy=0
類比: xy=F(x)微分1次: xy'+y=F'(x)微分2次: xy"+2y'=F"(x)=> F"(x)=-xy => y=-F"(x)/x=> y2=-sin(x)/x 6.解下列三個常數係數ODE(必須是實數解): (a) y"-y=0 0=D^2-1=(D+1)(D-1)D=+-1y(x)=c1*e^x+c2*e^(-x) (b) y"+y=00=D^2+1 => D=+-jy(x)=c1*cos(x)+c2*sin(-x) (c) y"+2y'+y=0 0=D^2+2D+1=(D+1)^2=> D=-1,-1y(x)=(c1+c2*x)e^(-x)


2014-11-02 19:34:02 補充：
7.解下列三個Euler-Cauchy方程(必須是實數解):

(a) x^2*y"-4xy'+4y=0

Set y=x^m => y'=mx^(m-1), y"=m(m-1)x^(m-2)

代入原式裡面:

0=m(m-1)-4m+4

=m^2-5m+4

=(m-1)(m-1)

m=1,4

y(x)=c1*x+c2*x^4



(b) x^2*y"-3xy'+4y=0

0=m(m-1)-3m+4

=m^2-4m+4

=(m-2)^2

y(x)=[c1+c2*ln(x)]*x^2

2014-11-02 19:34:27 補充：
(c) x^2*y"+xy'+4y=0

0=m(m-1)+m+4

=m^2+4

=0
m=+-2j

y(x)=c1*x^(2j)+c2*x^(-2j)




8.非齊次ODE: y"-4y'+3y=e^x

0=m(m-1)-4m+3

=m^2-5m+3

m=(5+-√13)/2

y(x)=c1*x^(5+√13)/2+c2*x^(5-√13)/2

2014-11-02 19:35:08 補充：
9.用Method of variation of parameters解非齊次ODE: y"+y=sec x

D^2+1=0 => D=+-j

yh(x)=c1*cos(x)+c2*sin(x)

yp(x)=A*cos(x)+B*sin(x)

w=|y1 .y2.|
..|y1' y2'|

=|.cos(x) sin(x)|
.|-sin(x) cos(x)|

=[cos(x)]^2+[sin(x)]^2

=1

A=-∫y2*sec(x)dx/w

=-∫sin(x)dx/cos(x)

=∫d(cos(x))/cos(x)

=ln[cos(x)]

2014-11-02 19:35:50 補充：
B=∫y1*sec(x)dx/w

=∫cos(x)sec(x)dx

=∫dx

=x

=> yp(x)=ln[cos(x)]*cos(x)+x*sin(x)

=> y(x)=yh(x)+yp(x)

=[c1*cos(x)+c2*sin(x)]+{ln[cos(x)]*cos(x)+x*sin(x)}

2014-11-02 19:36:32 補充：
請轉到意見欄

這里很不錯9９9ｓhｏpｐiｎg。cｏｍ老婆很喜歡
发乙

小貢獻一點：
次方的符號，是 ^

t的2次方=t^2