常微分方程练习题及答案(复习题)

1、常微分方程练习试卷xxe xebdx2y5y y 1d 2xydx (2x 3y 20)dy 0 yX A(t)X (t) A(t)xdt2xdyf(xy)y dx2y 2y y 0dy x y 1dx x y 3d2xx dt2xy x Oy(O) 1,y(0)2X * / 2xy y y e y (x) eW(t) 0x!(t),X2(t),川，Xn(t) a x-4y,-+4=8e-8y ysin x(cos xsin xxy)dxy(1x2)dy 0dXdtAX (t) dXAXdtx(0)dydx2x3y2 (1,0)(t),(x)(%3dx8y20(t) XA(t)X CXo, x

2、y(x) n(x),(x),(t)yo，Ax (t),(t)Cx 2y(Xo(x)(0)(x)xo,x ,尹（讪 卜吟）卩（小忡（Q X+pgy+g丁二opM 胃 0eO)吩)CO) rrW(t) ax (t。)(t) exp A(t t。)dt3,2,1u xy 1 du -dxu(f (u) 1) x3” Aty q i(t)exo)= iy(o)= o,y,(o)= oc1? +cax+c51 + 7522 A1f = -1 誥一 1= 0 / =-(-Da +c*U = i(x-1尸 +y=1dydx0，x0i,y 21,2,(1 z)dz1 z2-JAarctanz 3X(1 z2)

3、In | | Carcta nxy 2 In ,厂1厂厂2)21d2xx dt6k d7t=ydtR”空卩空十尸“dx dx ydy 宀 c dx cr+ v = U v 一 i,-. 0 drr J; x x 卯十 5(5 菱 5 “心护4加十4无二o血=0以站=2坯（0二U4G&阳心0北邛() = ai2+ 4(2就 +&) = Si 8口二 1二 0 k=J3G，5 C3x 二 j0()+ 工 *(/)-12 + c 斗 C2J + g沪y ysin xx/ Xy y y ce y c(x)ec(x)ex c(x)exc(x)ex sin x c(x)x e sin x1 xxc(x)

4、e x(sin x cosx) c y cex2-(sin x cosx)22 2(cos xs in x xy )dx y(1 x )dy2 2M (x, y) cosxsinx xy , N (x, y) y(1 x )2xycosxsin xdx(xy2dx yx2dy) ydy 0dsi n22x)1 2 2 1 2d(2xy) d(2y)0 sin2 xdXdtAX(t)dXdtAXx(0)det(AE)2)(5)0,2, 22,5V1,V20).(t)2t e2t e5te2e5t(0)(t)(t)1(0)2t e2t e5te2e5t2te 2e2te 4e5t5tdydx2x

5、13y2 (1,0)110(x)0i(x)yo:2x1 3 o(x)dxx,2(x)yoxi【2x13 i2 (x)dx 丄10x2x33 55x，(%3dx4xydy dx8y20dy 3 8y2dxx4ydy屯 pxdx dx32P 8y4yp2y(p3 4悄p(8y2p3)24y p(p3 4y2)(2ydp dy 2p c22pc2p) 02ydpcy(卫)2c2cx4c24(P)2cp34y214(4y2)3 y27 x321 x Ax (t), (0)1 4p()9 0 1,2 3 n11v(t)e3t2(A 3E)2i 0 i !n 1 . i e t-(Ai 0i!E)i1 3

6、t 1 t( 12)e2 2 t( 12)exp At e3t E t(A 3E)3t 103t 1 t0 11 1(t), (t)X A(t)XC (t) (t)C1(t)(t) X(t)1(t) (t)X(t)detX(t) 0(t)(t)X(t)(t)(t)X(t)(t)X (t) A(t) (t)X(t) (t)X (t)A(t)(t) (t)X (t)(t) A(t) (t)(t)X (t)0X (t) 0, X(t)(t)(t)C(x)(Xo, Xy(x)yoX x 2y(Xo)d ,Xo,x n(X)(X),，(X)(X)(X)yod ,o(x)yo,n(X)yo1()d ,Xo

7、,X (n1,2,)XoxXo(X)2o(x)|(|Xo)|l)dM (xXo), MmaxxX , l2|(X)|X|(X)1(X)|(2| (o()l)dXo)dXoM(X Xo)2,xmaxx2n|(X)1(X)|MLn1(x X。)(X)n(x)|2|n 1()l)dXL jMLX n!Xo(Xo)ndML(n 1)!n(X Xo)n 1(x)1(X)|k 1ML (xk!Xo)kk 1ML .k!()k n(x) XoX(X)(x)(x) Xoxr卫O)yl+p(x)y+q(x)y= 0卩(x)吩)pM审e材oo回时r . eh) 审S0 3 (R 卩仗)琢(x)工 0 (he (-叫+ro)心 w C-m-*3d)祝心)=”CG = o 叭码)=oeO)心 e (-cor-Ko)秋吗)=肌勺)=o w(x0) = 0 cpfx) g)心 Egg “ (心)二材 E) = 0PF(码)=0pw w(t)dX ax (to)(t) exp A(t t