四川師范大學2009至2010學年第一學期常微分方程期末考試試題B范文

1、四川師范大學2009至2010學年度第一學期常微分方程期末考試試題B四川師范大學數(shù)軟學院數(shù)學與應用數(shù)學期末考試2008 - 2009學年度第一學期期末考試TEST PAPER OF ORDINARY DIFFERENTIAL EQUATIONS (B)答卷說明：本試卷共6個大題，滿分100分，120分鐘完卷。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常微分方程試題B參考答案及評分細則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 ,尸立氣or尸如.15 pointsII . Solution .matrixThe characteristicequatio nConsider the coefficients of A isdet a種：) =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 (去際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常微分方程試題B參考答案及評分細則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 ,尸立氣or尸如.15 pointsII . Solution .matrixThe characteristicequatio nConsider the coefficients of A isdet a種：) =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 (去際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