[考題] 日/數學系/葉乃實/微分方程

[goshfju (cola)]

上學期期中考(答案全打要很久 所以只寫大方向喔)

1.Consider dy/dx= -(x^2)y , and answer the following questions:
(i)please draw the isoclines
(ii)please draw the direction field
(iii)attempt the approximate integral curves
(iv)in case y(0)=1, show that there is a unique solution
     by NOT solving the problem

(i)(ii)(iii)圖描仔細就好
(iv) f(x,y)=-(x^2)y
     ⊿f/⊿y=-x^2 is continuous near x=0
     by the uniqueness and existence thm.
     there is only one sol.

2.xy'(3(x^2)+2y)- 5y((x^2)+y) - (x^4) =0
  Please solve y(x).

f(x,y,y')=xy'(3(x^2)+2y)- 5y((x^2)+y) - (x^4)
f(tx,(t^m)y,(t^m-1)y')=(t^r)f(x,y,y')
could find it's a isobraic fn. with r=4 m=2
f(1,y/x^2,y/x^4)= f(x,y,y')/x^4 = (3/x)y' + (2/x^3)yy' - (5/x^2)y
                                   - 5(y^2/x^4) -1   ----(1)
let u=y/x^2 thus y=u(x^2) , y'=2ux+(x^2)u'       substitute into (1)
...
整理後得 xu'=(u^2-u+1)/(2u+3)
(1/x)dx = (2u+3)/(u^2-u+1) du
...
積分後得 ln|cx/(u^2-u+1)|=(8/√3)arctan((2u-1)/√3)
ln|(cx^5)/(y^2-(x^2)y+x^4)|=(8/√3)arctan((2(y/x^2)-1)/√3)

3.Find the orthogonal trajectories of the curves represented by the
  differential equation r^n=(a^n)sec(nθ) , where n is an positive integer
  and a is an arbitrary const.

r^n=(a^n)sec(nθ)
n(r^(n-1))r' = n(a^n)sec(nθ)tan(nθ)
tanΨf= r/r' = cot(nθ)
tanΨg= -1/tanΨf =-tan(nθ)= rg/rg'= rg/(drg/dθ)
drg/rg=-cot(nθ)dθ
ln|rg|= -(1/n)ln|sin(nθ)|+c
rg^n=csc(nθ) :orthogonal trajectories

4.y=y(x) and y'=(-1/x)+y/x+(y^2)/(x^3)
  please find the solution of y

y=x and y=-x are 2 PSs
y'=(-1/x)+y/x+(y^2)/(x^3)      ...(1)
x'=-1/x+x/x+(x^2)/(x^3)        ...(2)
(-x)'=-1/x+(-x/x)+(-x)^2/x^3   ...(3)
(1)-(2) (y-x)'=(y-x)/x + (y^2-x^2)/x^3
        (y-x)'/(y-x) = 1/x + (y+x)/x^3 ...(4)
(1)-(3) (y+x)'/(y+x) = 1/x + (y-x)/x^3 ...(5)
(4)-(5) (y-x)'/(y-x) - (y+x)'/(y+x) = 2/x^2
ln|(y-x)/c(y+x)|= -2/(x^2)
y= x[1+ce^(-2/x)]/[1-ce^(-2/x)]

5.(x^2+siny)y'+2xy=0 please solve y(x)

2xydx + (x^2+siny)dy
M=2xy N=x^2+siny
Mx=Ny=2x ...exact
   x           y
u=∫ 2tYodt + ∫ (Xo^2+sint)dt=c
   Xo          Yo
Let Xo=0 Yo=π/2
(x^2)y-cosy=c

就這樣啦@@""用BBS打數學式子真是噩夢啊
這是以前的題目..今年我沒把題目抄下來...不過也是大同小異啦
老師每年都會出像第一題的做圖題..
還有第二題arctan的積分會積出很醜的答案..今年也有一樣的..
其實期中考的題目都蠻基本的 跟講義裡的題目類似
期末考會比期中難非常多倍...想保住不被當的話..期中考高點吧@@""

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◆ From: 218.167.79.24
→ goshfju:今年的題目有一題也是求orthogonal trajectories
推 fm1210:真是太辛苦了 哇
→ goshfju:有一題是primitive=>D.E.
→ goshfju:其實是太窮了 來騙P幣的 XD
→ goshfju:有一題是bernoulli
→ goshfju:還有一題忘記了@@"想到再推
※ 編輯: goshfju         來自: 218.167.79.24        (01/24 23:33)