四川师范大学2009至2010学年第一学期常微分方程期末考试试题B范文

1、四川师范大学2009至2010学年度第一学期常微分方程期末考试试题B四川师范大学数软学院数学与应用数学期末考试2008 - 2009学年度第一学期期末考试TEST PAPER OF ORDINARY DIFFERENTIAL EQUATIONS (B)答卷说明：本试卷共6个大题，满分100分，120分钟完卷。Questio n nu mberIIIIIIIVVVItotalsig natu rescoresscoressig natureI . (15 points). Solve the first order implicit differential equationscoressig

2、natureII . (15 points) Solve the linear differential equationsscoressig natureIII . (15 points).Solve the differential equation(y2 -Sr2)血scoressig natureIV. (15 points) Solve the linear differential equation with the in itial-valuescoressig natureadiscuss of the and theV. (20 points) . If j 00/r0 a

3、nd stability of the trivial solution of the systemAc=-fee + aryy discoressig natureis a generalVI . (20 points) Prove that solution of the linear differential equation of first order常微分方程试题B参考答案及评分细则LetI . Solution .the n the ordi nary differe ntial equati on is rewritediF3-2jp+9jt=0(1),thenDiffere

4、ntiate the above equatio n with respect to xIt follows that)(p-x) = 0p-Gc 或 p-i35 points10 poi ntsFrom (1), we haveThe soluti ons of the ordi nary differe ntial equati on are9 C ,尸立气or尸如.15 pointsII . Solution .matrixThe characteristicequatio nConsider the coefficients of A isdet a种：) =0.2-1 1A=X =1

5、|.i 2-=a-i)u-3)=oThe matrix A has three eige nv alues are5 poi ntsWhen Afrom (去际0,we findWhen ,from , we findU丿The fun dame ntal matrix soluti on of10 poi ntsis哄)=& /丿Then,The particular solution of the equation isdrifThe solution of linear ordinary differential equations isfj=4&, we haveare negativ

6、e real , andThe characteristic equation isThe eige nv alues are6poi nts1)If 匸 A 曲,we have 兀為 are positive , real , and.The n the point (0,0) is an un stable improper node.The n the point (0,0) is an stable improper node.4)c03a c 00/r0 a nd stability of the trivial solution of the systemAc=-fee + ary

7、y discoressig natureis a generalVI . (20 points) Prove that solution of the linear differential equation of first order常微分方程试题B参考答案及评分细则LetI . Solution .the n the ordi nary differe ntial equati on is rewritediF3-2jp+9jt=0(1),thenDiffere ntiate the above equatio n with respect to xIt follows that)(p-

8、x) = 0p-Gc 或 p-i35 points10 poi ntsFrom (1), we haveThe soluti ons of the ordi nary differe ntial equati on are9 C ,尸立气or尸如.15 pointsII . Solution .matrixThe characteristicequatio nConsider the coefficients of A isdet a种：) =0.2-1 1A=X =1|.i 2-=a-i)u-3)=oThe matrix A has three eige nv alues are5 poi

9、ntsWhen Afrom (去际0,we findWhen ,from , we findU丿The fun dame ntal matrix soluti on of10 poi ntsis哄)=& /丿Then,The particular solution of the equation isdrifThe solution of linear ordinary differential equations isfj=4&, we haveare negative real , andThe characteristic equation isThe eige nv alues are

10、6poi nts1)If 匸 A 曲,we have 兀為 are positive , real , and.The n the point (0,0) is an un stable improper node.The n the point (0,0) is an stable improper node.4)c03a c 46, we haveare conjugative imag inarynu mbers , and real of them are n egative . Then the point (0,0) is an stable spiral point.14po i

11、ntsIfThen, the trivial solution of system (1) is stable ifand thetrivial solution of system (1) is unstable ifWe have that the trivial solution ofthe orig inalsystem is stable if匸Q and the trivialsoluti on of theorig inalsystemis unstable if 匸: 020 poi ntsVI. Proof.The associated homogeneousequation of the equation is which has the soluti onWe assume eq