MATH2010 ASSIGNMENT 2, SEMESTER 1, 2020 . Q1) Consider the ODE y"(t)- 6y'(t) - 3y(t) = 0, (1) where y'(t), y"(t) denote dh and dz respectively. (a) Find the roots of the characteristic equation corresponding to (1) and use these to write the general solution y(t) to the ODE (1). (2 marks) Sol. The characteristic equation is A2 - 6A - 3 = 0. This has the two real solutions A = 3+2v3. The general solution to (1) is y(t) = ae(3+2/3)t + be (3-2/3)t. (b) Couple the ODE (1) so that it is expressed in the form Y'= AY + b, (2) 0 0 (herarK's)= z y y , A is a 2 x 2 matrix, and b is a vector. Sol. Let z = y'. Then y' = z and y" = z' = 3y + 6z. Putting these -- together, = 0 0 1 Y Y (c) Find the eigenvalues and eigenvectors of A and use these to plot the phase portrait for the system (2). (2 marks) Sol. We found A = 0 0 1 det(A- AI2) = 3 6 . 0 -1 1 2 -613-3,6 -1 , 0 = 1-3-23 1-3+2/3. The eigenvalues of A are \1 = 3+2v3 and /2 = 3 - 2v3. We have 0 -3-213 1 0 - A- >1I2 = 3 3-2v3 ? 0 The first eigenvector is y(1) = 1 -3-2v3 1 0 - 0 1 0 0 : 3+2/3 - A - A212 = 0 -3+2v3 3 3+2v3 Key words and phrases. . . Also, _ -3+2v3 1 0 ? 0 0 : 1
2 . Using the Wolfram - 0 1 3 -21/3 Q = StreamPlot[ {z, 3 y + 6 z}, {y, -3, 3}, {z, -3, 3}, AspectRatio -> Automatic] Export["phaseport1.eps", Q] we get: 3 2 1 0 -1 -2 -3 -3 -2 -1 0 1 2 3 (d) Does the system (2) obey the superposition principle? Explain. (2 marks) Sol. Yes. The system is linear and homogeneous. Q2) Consider the ODE - 4y"(t) + 3y'(t) + 5ty(t) = cos(t). (3) (a) Show that this ODE (3) is linear but inhomogeneous. (2 marks) Sol. The ODE is linear because all derivatives of y appear as linear function of themselves with coefficients which are functions of t. To show the ODE is inhomogeneous, suppose y(t) is a solution. Assume cy(t) is also a solution for all c E R. Then -4cy"(t)+3cy'(t)+5cty(t) = c(-4y"(t)+3y'(t)+5ty(t)), = c cos(t). but this is only a solution when c = 1, a contradiction since we assumed cy(t) is also a solution for all c E R. (b) Couple the ODE so that it is expressed in the form (4) pzhfr&rÄs)= z Y' = AY + b, y = y 0 y' , A is a 2 x 2 matrix, and b is a vector.
MATH2010 ASSIGNMENT 2, SEMESTER 1, 2020 3 Sol. In matrix form, letting z = y' and Y = 0 5t 3 0 0 0 HP Again, this is