Math 331 - ODE Written Problem Set # 3
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hw3.pdf
Math 331 - ODE Due: 11 or 12 December, 2018 Written Problem Set # 3
Attempt and turn in all problems.
1. Write each of the following differential equations as a first order system. Is the system linear or nonlinear?
(a) y′′′ + 4t2y′′ −y′ = 0 (b) y′′ + y′ −y2 = 0 (c) y′′′′ − 2y = 5e−t
2. For each of the following matrices, compute the eigenvalues and eigenvectors. Also indi- cate the type of the critical point of the system (saddle, source or sink, spiral, etc.)
(a) A =
( 2 −1 −1 2
) ; (b) B =
( 0 −2 2 0
) ; (c) C =
( 3 −4 1 3
) .
3. Do the following computations.
(a) Let
A =
( 1 + i −1 + 2i 3 + 2i 2 − i
) and B =
( i 3 2 −2i
) ,
and compute the following: (i) A− 2B; (ii) 3A + B; (iii) AB; (iv) BA; (v) A∗; (vi) B∗.
(b) Let
x = (
2, 3i, 1 − i )T
and y = ( −1 + i, 2, 3 − i
)T ,
and compute the following, showing details: (i) xT y; (ii) yT y; (iii) (x,y); (iv) (y,y).
4. Consider the system
x′ = Ax, with A =
( 2 −5 1 −2
) ,
and the two vector-valued functions
x1(t) =
( 5 cos t
2 cos t + sin t
) and x2(t) =
( 5 sin t
2 sin t− cos t
) ,
(a) Show that both x1 and x2 are solutions of the system.
(b) Show that for any constants c1 and c2, x = c1 x1 + c2 x2 is also a solution.
(c) Find the Wronskian W [x1,x2](t).
(d) Show that the pair (x1, x2) forms a fundamental set of solutions of the given system.
(e) Find the solution that satisfies the initial condition x(0) = (1, 2)T .
5. Solve the initial value problem
x′ =
( 5 −1 3 1
) x, x(0) =
( 2 −1
) ,
and describe the behavior of the solution as t →∞.
6. Solve the initial value problem
x′ =
( −2 1 −5 4
) x, x(0) =
( 1 3
) ,
and describe the behavior of the solution as t →∞.
7. Solve the initial value problem
x′ =
( 1 −5 1 −3
) x, x(0) =
( 1 1
) ,
and describe the behavior of the solution as t →∞.
8. Consider the system with (real) parameter α,
x′ =
( α 1 −1 α
) x.
(a) Determine the eigenvalues in terms of α.
(b) Find the bifurcation value or values of α; these are the values at which the qualitative nature of solutions change.
(c) For each such bifurcation value αc, sketch a phase portrait for values of α slightly below, equal to and slightly above αc. Be sure to indicate the direction of flow.
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