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Instructions: If your homework is not already in electronic format, please scan it into one PDF file, and
upload it on GradeScope. Possible apps for scanning your work are CamScanner or GeniusScan: if you
believe that turning your document into Black & White (not grayscale!) makes it more readable, please
do so (also: writing in pen instead of pencil may make the scanned document more readable). GradeScope
makes it possible to ‘tag’ the individual problems within your PDF file: please do it in order to make the
graders’ work a bit simpler. Please solve the problems in the given order, and show your work.
Problem 13.1. Suppose that yp, (x) is a particular solution to Lly] = b1(x), and that yp.(x) is a particular
solution to L[y] = b2(x) (for the same operator L: i.e. they are the same equation but with different 6’s).
Using the linearity of L show that yp, (x) + yp. (x) is a solution to L[y] = bi) (x) + be(x).
Problem 13.2. Find a particular solution to the following differential equations:

(a) y” + 16y = e** (b) y” —y' —-2y=3r+4 (c) y” —y — by = 2sin(3z)
(d) 4y" + 4y' + y = 32xe" (e) x” + 2y' — 3y = 14 xe* (f) "+ 4y' = 32-1
Problem 13.3. Find the general solution y(x) = ye(x) + yp(x) to the D.E.: y” + y' = 2 — sin(z).
Problem 13.4. Find the general solution of the D.E. y”+y = sin*(z) using, for yp: (i) the method of varia-
tion of parameters, and (ii) the method of undetermined coefficients (tricky; what is your ‘guess’ for yp(x)?).
Problem 13.5. We will prove the direct implication (“=>”) of Theorem 2 in Lecture 35. So, assume that
yi(x) and y2(x) are two linearly independent solutions of an homogeneous DE ao(x)y"+a,(x)y’+ao(x)y = 0.
Let’s define z1 (x) = ae , Z2(x2) = a , the Wronskian W (x) = det([zi(2) z2(x)]). We want to prove
1 2
that W(x) 4 0 for all x. Proceeding by contradiction, we assume that there exists x9 such that W (xo) = 0.
(a) Deduce that there exists a scalar \ such that z1(2o9) = Azo(xo9) or Zo(x%o) = AZ1 (Xo). (Without loss of
generality, let us assume that z1(2%9) = Az2(xo0).) (b) Justify that the function y(x) = Aye2(x) is a solution
of the D.E. a2(x)y” + a1(x)y’ + ao(x)y = 0, and that it satisfies the same initial condition at xg as y1(2).
(c) Using the uniqueness of solutions of an I.V.P. (that still holds for higher order D.E.s under appropriate
conditions, which we assume hold here), deduce that y(x) = y1(x) for all x, and obtain the contradiction.
Problem 13.6 (nilpotent matrices). A (n x n) matrix A is called nilpotent of order k if AX = O but
A’ #£O for 1 < €< k—1 (for example, a nonzero matrix A is nilpotent of order 2 if A? = O). Suppose
that the (n x n) matrix A is nilpotent of order k. Show that: (a) A is singular. (b) Its only eigenvalue
is \=0. (c) (In — A) is invertible, with inverse B = I, + A+ A2+...+ A*7!. Hint: Compute (I, — A)B.
Problem 13.7. Consider the generic 2-dimensional, first-order, linear, homogeneous, constant-coefficients
dynamical system (D.S8.): r
» » ( au + Oy or simply: x’ = Ax, where x = “land A-| ° ° ,
y =cxr+dy y c d
Show that: 2” — Tr(A)a’ + det(A)x = 0 and y” — Tr(A)y/ + det(A)y = 0. (Remember: Tr(A) = a+ ad).
r
Problem 13.8. Consider the dynamical system (D.S.): y _ au with I.C.: 2(0) = 1, y(0) = 3.
(a) Solve it ‘by hand’, i.e. not resorting to the exponential matrix: first find a D.E. for x(t), solve it, and
then employ the first of the two equations in the D.S. to find y(t). Use the initial conditions (I.C.).
(b) The solution (x(t), y(t)) gives a parametric curve on the xy-plane. We can ‘climinate’ the parameter t
and identify the curve. Compute the square of each of x(t) and y(t), and show that y?(t) — 2x*(t) = 7 for
all t € R. So the solution lies on the curve y* — 2x” = 7: do you recognize this curve?

Problem 13.9. Consider the dynamical system: ‘ ¥ 6 with initial conditions x(0) = x0, y(0) = yo.
(a) Solve it ‘by hand’, by first finding y(t) and then x(t), in terms of zo and yo. (b) Write the D.S. in the
form x’ = Ax. Show that A is nilpotent, and find the solution again by computing: x(t) = e4*x(0).
,
Problem 13.10. Consider the dynamical system: ‘ ¥ _ " with initial conditions x(0) = x0, y(0) = yo.
Write it in the form x’ = Bx, show that B is nilpotent, and find the solution by computing: x(t) = e?*x(0).
(In this problem, solving the dynamical system ‘by hand’ is not required.)
,
Problem 13.11. Consider the dynamical system: ‘ ¥ _ * with initial conditions x(0) = x0, y(0) = yo.
(a) Solve it ‘by hand’ (we did a similar example in class). (b) Now write it in the form x’ = Cx.
Unfortunately, this time the matrix C is not nilpotent. However, show that: C! = C, C? = —In, C? = —C,
C+ = In, C° = C, et cetera. The matrix sequence C* is periodic with period 4, i.e. C**4 = C*, for all k.
We also have C?* = (—1)* Ig, and C#*+! = (—1)*C. (c) (Challenging!) Show that we can write:
Ct . _ | cos(t) — sin(t)
e~' = cos(t)I + sin(t)C = | —sin(t) cos(t) |’
(d) Finally, write explicitly the solution: x(t) = e©'x(0). Hint: For (c), we can split the sum of a
convergent series as the sum of its even-indexed terms and the sum of its odd-indexed terms. L[.e., in
formulas: SoP° 5 ak = Sop. Gak + Dopp Gak+1- You will also need the the Taylor series for cosine and sine:
— (=1)F 94 — (=1)* opt
cos(t) = ) | Gaye > Sin) » (2k + 1)!
k=0 k=0
Theorem. Let A and B be (nxn) matrices. If AB = BA, then e4+? = e4e?.
Problem 13.12. In general, it is not the case that e(4+®)* = e4'e®*, Find two (2 x 2) matrices A and B
such that AB £ BA and e(4+8)t F e4teBt. Hint: Look at problems 13.9, 13.10, and 13.11.
Problem 13.13. Consider the system of three linear differential equations: v= 2%, + 322 + 4x3
where the unknowns are the three functions x;(t), r2(t), and x3(t). ry = 2x2 + 6x3
(a) Write the system in the form x’ = Ax, where A is a (3 x 3) matrix. Ly = 273
(b) Write A as the sum of two matrices, A = D+U, where D is a diagonal matrix (all of the off-diagonal
entries are zero, and the diagonal entries are exactly the diagonal entries of A), and U is a ‘strictly’ upper
triangular matrix (all of the entries on or below the main diagonal are zero). Verify that DU = UD.
(c) Find: e4¢ = e?teY* (why it this formula correct?). Note that U is nilpotent of order 3. (d) Finally,
find the solution to the dynamical system above, with initial conditions 2;(0) = 3, x2(0) = 6, 73(0) = 4.
Problem 13.14 (Properties of e4). Besides the Theorem cited above, the matrix exponential has several
properties, helpful for solving dynamical systems. Using the definition of matrix exponential, show that:
(a) e? = I,, where O is the zero matrix.
(b) Se“! = Ae“. Hint: In the definition of e4’ = I+ At+5(At)?+ (At)? +..., note that (At)* = A*t*,
then differentiate the infinite sum (term by term) treating A* as a constant.
(c) (e4)—! = eA (so, the matrix e4 is always invertible). Hint: Use the theorem above (with B = 2).
(d) If S is a (n x n) nonsingular matrix, and B is any (n x n) matrix, then eS BS — SeB S71.
(e) If \ is an eigenvalue of A, then e? is an eigenvalue of e4.
Hint: Show that if Ax = Ax for some x # 0, then e4x = ex.
Problem 13.15. Property (d) of Problem 13.14 is extremely helpful when the matrix A in the linear
system x’ = Ax is diagonalizable! In this case we can write A = SDS, and by the previous problem,
cAt — -SDS~'t _ ,S(Dt)S~* _ SePtg-1.
where S is a nonsingular matrix whose columns are the eigenvectors of A, and D is a diagonal matrix
whose diagonal elements are the eigenvalues of A. Since D is diagonal, e”* is very simple to compute!
Following the above procedure, solve the following two linear systems:

war 3 5 v= @1+2%24+ 223 1
(a) : , 9% | 9 with: x(0) = | (b) 4 25 = 227, + 22+2x3 with: x(0) = | 3].
y= J Ly = 241 + 242+ L3 2
For (b), you may use the solution to Problem 10.16 in HW#10.
The next problems, on the Laplace Transform (Lectures 87 and 38), are for extra credit (20 points total).
Problem 13.16. In Problem 11.3 of HW#11, by computing an integral you found: C{sin(bt)} = 2am:
s
Now, without computing any integral, find the Laplace transforms of the following functions:
(a) fi(t) = e@ sin(dt) , (b) fo(t) = te® sin(bt) .
Hint: use the properties of the Laplace transforms (Theorems 1, 2, and 3 of Lecture 38).
Problem 13.17. A function that is very useful J ys Walt)
in engineering is the so-called Heaviside function: 4 1 L
ua(t) = 0 for0O<t<a nn ~~
ow’) 1 for t>a Ye
where a > 0 is a constant. It is plotted on the right. as
(a) Show that its Laplace transforn is Ug(s) = —[ ZA0 Q, +,
(b) For a generic fucntion f(t), its shifted version by a constant a > 0 is defined as follows:
fa(t) = f(t — a)ug(t) (note that fa(a) = f(0)). This is illustrated in the figure below.
y S f
= Z = t-aju te
Zz y2 4(b) | Zoo ( ) af )
xo shift |
~
Li en !
A ,
40 t G0 Ww ——— t
QZ
Z
Assuming that the Laplace transform of f(t) is F(s) = L{f(t)}, show that L{fa(t)} = e~* F(s).
(c) Use the above properties to find, without computing

an integral, the Laplace transform of one period of the y Y _ (&)

function sin(ct) (where c > 0 is a constant), namely ZZ 4 ~ 3.

of the function: . Zz

sin(c(t—a)) fora<t<b reer
g(t) = : 0 otherwise 0 \ / L t
where a > 0 is a constant and b = a+2z7/c. The func-

tion y = g(t) is illustrated on the right. Hint: express

g(t) as the difference of two shifted sine fuctions.

Problem 13.18. Use the method of Laplace transforms to solve the following Initial Value Problem (IVP):
D.E.: y” + y = cos(2t) LC.: y(0) =0, y/(0) = 1.
Problem 13.19. Consider the following Initial Value Problem (IVP):

. fa =x2-3yt+e*% we we xr(0) =0
Dynamical System (D.S.): : y! = 2x — Ay ; Initial Conditions (I.C.): : y(0) <1
We denote by X(s) and Y(s) the Laplace transforms of x(t) and y(t) respectively.

1
(a) Show that we have: (s — DX(s) + 8¥(s) = s+4
—2X(s)+(s+4)Y(s) =1
(b) Find the Laplace transforms X(s) and Y(s). (c) Find the solution (x(t), y(t)) of the I.V.P.