Solutions for homework Section 9 Linear Sstems with constant coefficients Overview of the Technique 3 Use hand calculations to find the characteristic polnomial and eigenvalues for the matrix ( 3 5 λ T r(aλ + det(a λ + 7λ + λ λ 5 real distinct 5 Use hand calculations to find the characteristic polnomial and eigenvalues for the matrix ( 5 3 6 4 λ T r(aλ + det(a λ λ λ λ real distinct 7 Use hand calculations to find a fundamental set of solutions for the sstem A where A is the matrix ( 6 8 λ T r(aλ + det(a λ 4λ λ λ 6 real distinct For the eigenvalue λ the eigenspace is the nullspace of A + I ( 6 + 8 + ( 8 8
( which is generated b the vector v For the eigenvalue λ 6 the eigenspace is the nullspace of ( 6 6 8 A 6I 6 ( which is generated b the vector v Therefore a fundamental set of solutions is (t e t ( ( 8 8 ( and (t e 6t 9 Use hand calculations to find a fundamental set of solutions for the sstem A where A is the matrix ( real repeated Therefore a fundamental set of solutions is hence for an nonzero vector λ T r(aλ + det(a λ + λ + λ λ (t e t and (t te t ( + + (t e t Note that the given sstem is a decoupled sstem of DE { (
Section 9 Linear sstems with constant coefficients: Planar sstems 3 Find the general solution of the sstem A ( 5 λ T r(aλ + det(a λ + 7λ + λ 3 λ 4 real distinct For the eigenvalue λ 3 the eigenspace is the nullspace of ( 5 + 3 A + 3I + 3 ( which is generated b the vector v ( For the eigenvalue λ 4 the eigenspace is the nullspace of ( ( 5 + 4 A + 4I + 4 ( which is generated b the vector v Therefore the general solution is ( ( (t e 3t + e 4t 3 The complex values vector z(t is given Find the real and imaginar parts of z(t ( z(t e it + i Solution Using Euler s formula we have ( + i z(t e (+it + i [( e t (cos t + i sin t { ( ( } cos t sin t + i ( + i ] { cos t ( + sin t ( } therefore the real part is Re ( z(t ( ( ( cos t cos t sin t cos t sin t 3
and the imaginar part is Im ( z(t ( ( ( sin t cos t + sin t cos t + sin t 5 The sstem ( 3 3 ( 6 3 has complex solution z(t e 3it ( i Verif b direct substitution that the real and the imaginar parts of this solution are solutions of sstem ( Then use Proposition 5 in Section 85 to verif that the are linearl independent solutions Solution Using Euler s formula we have ( i z(t e (+3it + i [( ( ] therefore the real part is and the imaginar part is (cos 3t + i sin 3t { cos 3t (t cos 3t (t cos 3t ( sin 3t + i ( + i } ( sin 3t ( + sin 3t ( ( { ( ( cos 3t + sin 3t } ( cos 3t + sin 3t cos 3t ( cos 3t sin 3t sin 3t We have that and (t (t ( cos 3t + sin 3t cos 3t ( cos 3t sin 3t sin 3t ( 3 sin 3t + 3 cos 3t 6 sin 3t ( 3 sin 3t 3 cos 3t 6 cos 3t On the other hand ( ( ( 3 3 3 3 cos 3t + sin 3t 6 3 (t 6 3 cos 3t and respectivel ( ( ( 3 3 3 3 cos 3t sin 3t 6 3 (t 6 3 sin 3t 4 ( 3 sin 3t + 3 cos 3t 6 sin 3t ( 3 sin 3t 3 cos 3t 6 cos 3t
which shows that the real and imaginar parts are solutions of the given sstem To check the linear independence of (t (t it is sufficient to check it at one point At t ( ( ( and ( which are linearl independent: det ( 59 Figure 8 shows two tanks each containing 36 liters of salt solution Pure water pours into tank A at a rate of 5L/min There are two pipes connecting tank A to tank B The first pumps salt from tank B into tank A at a rate of 4L/min The second pumps salt solution from tank A into tank B at a rate of 9L/min Finall there is a drain on tank B from which salt solution drains at a rate of 5L/min Thus each tank maintains a constant volume of 36 liters of salt solution Initiall there are 6 kg of salt present in tank A but tank B contains pure water (a Set up in matrix-vector form an initial value problem that models the salt content in each tank over time (b Find the eigenvalues and eigenvectors of the coefficient matrix in part (a the find the general solution in vector form Find the solution that satisfies the initial conditions posed in part (a (c Plot each component of our solution in part (b over a period of four time constants (see Section 47 or Section Exercise 9 [ 4T c ] what is the eventual salt content in each tank? Wh? Give both a phsical and mathematical reason for our answer Solution (a Let x(t salt in st tank (t ( salt in nd ( tank x( 6 The initial conditions are then and the equations write ( hence (b The characteristic equation is d dt x 36 9 9 (5 + 4 36 ( x ( 4 dx dt 9 4 4 36 4 x 36 9 (x λ λ Tr(A + det(a λ λ( 4 + ( 6 36 λ + λ( + 5 44 hence the eigenvalues/eigenvectors are λ ( 4 v 3 λ ( v 3 5
The general solution is ( ( (t c e t 4 + c 3 e t 3 The solution corresponding to the initial condition is obtained from ( ( ( 6 c + c 3 3 ie c c 5 hence ( ( x(t 5e t 4 + 5e t (t 3 ( ( 3e t 4 + 3e t 3 45e t 4 + 45e t (c T c 4 T c therefore we plot the solution on [ 4T c ] [ 48] 6 x(t (t 5 4 3 5 5 5 3 35 4 45 5 6
4 Find the general solution of the sstem A for the matrix ( 4 6 λ T r(aλ + det(a λ 8λ + 6 λ λ 4 real repeated The general solution is (t e 4t + te 4t (A ( 4I 4 e 4t + te 4t for an vector such that (t 49 Find the solution of the initial value problem for sstem A with matrix and the initial value ( Solution The solution is ( 3 ( 4 6 ( 4 (t e 4t + te 4t ( ( ( 3 4 3 e 4t + te 4t ( ( 3 e 4t + te 4t ( 3 t e 4t t 7
3 Section 93 Phase plane portraits Calculate the eigenvalues to determine whether the equilibrium point is a spiral sink or a source Calculate and sketch the vector generated b the right-hand side of the sstem at the point ( Use this to help sketch the trajector for the sstem passing through the point ( Draw arrows on the solution indicating the direction of motion Use our numerical solver to check our result ( 3 6 3 5 Solution The characteristic equation is λ λtr(a + det(a λ λ + hence the eigenvalues are λ ± i therefore ( the origin is a spiral source At : ( ( 3 6 3 5 ( 5 giving a clockwise motion x -x+ -5x+3 5 5-5 - -5 - -5-3 -35-4 - -5 - -5 5 5 5 3 35 4 x 8
4 Section 6 Numerical methods: Euler s method 3 Consider the initial value problem t ( Hand-calculate the first five iterations of Euler s method with step size h Arrange the result in a tabular form k t k k f(t k k h f(t k k h 3 3 3 39 39 4 4 6 444 444 5 5 36 558 557 5 Consider the initial value problem z x z z( Hand-calculate the first five iterations of Euler s method with step size h Arrange the result in a tabular form k x k z k f(x k z k h f(x k z k h - 8 5-5 65-3 3 54 78-78 4 4 46 54-54 5 5 496 39-39 9