常微分期终考试试卷(1)一、填空题（30%）x的积分因子的充要条件是有只含1、方程0y)dy?,x,y)dx?N(xM(（）。有只含的积分因子的充要条件是______________。y２、_____________称为黎卡提方程，它有积分因子______________。３、__________________称为伯努利方程，它有积分因子_________。nn个解，则它们线性４、若为阶齐线性方程的)),,Xt(),X(tX(tn12无关的充要条件是__________________________。５、形如___________________的方程称为欧拉方程。'????x(t)xA?(t))(t)(t)t(具有的关的基解矩阵，则和６、若和都是系是_____________________________。７、当方程的特征根为两个共轭虚根是，则当其实部为_________时，零解是稳定的，对应的奇点称为___________。二、解答题（６０％）30dy?yx?ydx(?)1、??２、tcos2t?xsin??x?21????1????并求的解３、若试求方程组Axx???(?At),(0)?????4?1????2expAtdydy23４、0y?8(?)4xy?dxdxdy2经过（0，0５、求方程）的第三次近似解y?x?dxdxdy6.求的奇点,并判断奇点的类型及稳定性.5??1,?x?y??x?ydtdt三、证明题（１０％）nn个线性无关解。１、阶齐线性方程一定存在试卷
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八年级地理上册填图题岩土工程勘察试题省略号的作用及举例应急救援安全知识车间5s试题及答案
一填空题?M?N?M?N??x??yx??y??１、)(?y)(?xM?Ndy2yz?y?２、)x())p?(xyx?Q(y?Rdxdy?dx)(x?(n1)pn?ne),(uxy?y３、)yx(p?)(Q?xydx４、0t)]?t),,x(w[x(t),x(n211n?ndyyddn?ay???xa?a0５、nn1?11nn?dxdxdx??６、Ct)?)((t７、零稳定中心二计算题?M?N?1,??1?x?y，所以此方程不是恰当方程，方程有积１、解：因为23?dy1yx?dx12yln?y??，两边同乘分因子得?0dy???ee(y)?222yyyyx?????31?y?xy??所以解为c??dydx???2y?yy??????2yx2c???cy)x?y(2另外y=0也是解即2y2????故特征根线性方程的特征方程２、??i0xx??0?1??x?t(Acost?Bsint)i?原方程有特解是特征单根，tsin?f(t)11?不是特征B=0A=-代入原方程i?2tcos2f(t)??221根，原方程有特解代入原方程B=0t?tBsin2?xAcos2?A311所以原方程的解为x?ccost?csint?tcost?cos2t2123??2?12????此时３、解解：得0?9?6?p(?)?3?2,1??14k=12n?1i?????1??)(???t??????t12111?t33ti??v??(A?3E)?e)?e(t?????????????)???t(!i????????222210i?i1?nt??it?得由公式expAt=)Ee(A?!i0?i?10?11?1?tt????????t3tt33?e?)t?eE?t(A?3expAt?eE????????01?11?t1?t????????3dy??2?8y??32y?p8dydx??x??x（则有４、解：方程可化为*）令p?dy4ypdx4ydxdp23223求导：y（*）两边对py??p4y4)(8?py)2y(p?dy1dpdpp3222cyp?y即将由得即0?0?2y?(pp?4y)(2y?p))?(ycdydy2?2pcx???2p2c?24c??x含参数形式的通解为：即方程的（p代入*）?24cp?2)y?(?c?为参数14223330p?4y?)p(4?y也是方程的解得）得：又由*代入（xy?27?0?y?002xx?????yxdx0120５、解：522xxxx???)dx??y?(x?0220420825114107xxxxxxxx?????(x???)dx?y??03160204400440020200??1?x?y?则）令X=x-3,Y=y+2６、解：由解得奇点（3，-2?05?x?y??dx?yx?????dt?dy?y?x??dt?1??1?）0故有唯一零解（0，0因为=1+111??1?122?????）-2故（3，得由i??1?0???2??1?1?2?2?1?1?为稳定焦点。三、证明题阶齐线性方程一定存在满足如下条件的n由解的存在唯一性定理知：解：n0,?(t)?1,xt)(x(,xt)?00001n2''1,0,?x(t)?)(,xt?0x(t)0020n11n??n?1n11?,x(t?(?(xt)0,xt))0,01002n010010考虑?(x,),(x),(01??w[xttt)]010n20010从而是线性无关的。)1it(x)(?,2,ni
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