Université Paris-Dauphine Differential equations: exercises L3 Maths, 2020-2021

Part 2: systems of linear differential equations. Weeks 7, 8, 9

Week 7: exercises 42, 43, 44. Week 8: exercises 47 and 52/53. Week 9: 54 /55 (56).
Of course, time allowing, feel free to cover more exercises.

A short revision to refresh students memory on change of coordinates and diagonalization of a matrix.
Change of coordinates. Let f be a linear map from Rd to Rd . Let B and B̃ be basis of Rd , and
A and à be the matrices of f in the basis B and B̃, respectively. Then

A = P −1 ÃP

where P is an invertible matrix such that:

- the columns of P give the coordinates of the vectors of B in the basis B̃
- the columns of P −1 give the coordinates of the vectors of B̃ in the basis B.
   
1 1
Thus, if d = 2, B = (e1 , e2 ) is the canonical basis, and B̃ = (η1 , η2 ) with η1 = , η2 = ,
    0 1
−1 1 1 1 −1
then P = and P = .
0 1 0 1
The second column of P expresses the fact that e2 = −1.η1 + 1.η2 .
A way to remember which matrix is P and which is P −1 is to start from the relation

AX = P −1 ÃP X

where X is a column vector. This relation means that if I am given a vector ~v with coordinates X in
the basis B and want to compute the coordinates of f (~v ) in the same basis, I can do it in two ways.
First, I can multiply X by the matrix of f in the basis B, that is, by A. Alternatively, I can find the
coordinates Y of ~v in the basis B̃, compute the effect of f in B̃ (that is, multiply by Ã), and then
go back to B.
Thus, the effect of the matrix P must be to transform the coordinates
 X of ~v in the basis B into
1
its coordinates Y = P X in the basis B̃. In particular, P , which is the first column of P ,
0
must give the coordinates of the first vector of B in the basis B̃. More generally, P gives the co-
ordinates of the vectors of B in the basis B̃, and P −1 the coordinates of the vectors of B̃ in the basis B.

Diagonalization. To diagonalize the matrix A ∈ Md (R), we first find the eigenvalues λ1 ,.., λk
by solving det(A − λI) = 0. We then find the eigenvectors by solving (A − λi I)X = 0 for each
i ∈ {1, .., k}. If A has d distinct eigenvalues, or more generally, if for each eigenvalue λi , the dimension
of Ker(A − λi I) is equal to the multiplicity of λi as a root of the characteristic polynomial of A, then
A is diagonalizable. An eigenvector basis B̃ is then obtained by grouping basis of Ker(A − λi I) for
i ∈ {1, .., k}. Denoting by P −1 the matrix whose columns give the coordinates of the vectors of the
eigenvector basis in the canonical basis, we then have A = P −1 DP with D diagonal.
Recall that when the matrix A is triangular, then it is very easy to find its eigenvalues. Why is that
so? Can you prove it?

1
Exercise 42 (explicit solutions and phase portrait of a saddle) Give the solutions and the phase
portrait for: a) x0 = −x, y 0 = y ; b) x0 = x + 3y, y 0 = x − y.
For b), give the matrix A, its eigenvalues and eigenvectors, show that A is diagonalizable in R,
give the solutions of X 0 = AX associated to the eigenvectors, and the general solution of X 0 = AX
through the coordinates of the solutions in an eigenvector basis. Note that if we only want to draw the
shape of the phase portrait, we just need to compute the eigenvalues and the associated eigenvectors.

Exercise 43 (a diagonalizable
 matrix in C ) Give the solutions and the phase portrait of X 0 = AX
a b
for A = with b > 0. Distinguish the cases a > 0, a = 0 and a < 0.
−b a

Exercise 44 (phase portraits of a source in different basis)

 
1 0
Let D = . Give the phase portrait of X 0 = P −1 DP X for :
0 2
       
−1 −1 0 1 −1 0 −1 −1 2 1 −1 1 2
a) P = I; b) P = ; c) P = ; d) P = ; e) P = .
1 0 1 0 1 2 2 1

Exercise 45 Find the solutions and sketch the phase portraits of X 0 = AX for:
   
0 0 0 −1
a) A = ; b) A = .
0 −1 0 0

Exercise 46 (trace-determinant plane and phase portrait) Let A ∈ M2 (R). What can be said of
the shape of the phase portraits of X 0 = AX when: 1) detA < 0 ; 2) T r(A) > 0 ; 3) det(A) > 0 and
T r(A) < 0 ; 4) detA = 0 and T r(A) < 0.

Exercise 47 (exponential of a matrix, paying attention to the condition that D and N commute)
Let t be a real number. Compute eA and etA for the following matrices:
     
2 0 1 2 1 1
i) A1 = ; ii) A2 = ; iii) A3 = ;
0 −3 0 1 0 2
 
0 0
In each case, give the solution of X = Ai X such that X(0) = .
1

Exercise 48 (comparison of three methods) Let x0 and y0 be real numbers. Solve the following
system of differential equations (H) and find the solution with initial condition (x0 , y0 ) at t = 0.
 0
x (t) = x(t) + y(t)
(H)
y 0 (t) = 2y(t)

Same question for the system (NH):

x0 (t) = x(t) + y(t) + et


(N H)
y 0 (t) = 2y(t) − 3e−t

Solve (NH) in three ways: i) using the fact that the system is triangular; ii) finding a basis
(X1 (·), X2 (·)) of the set of solutions of (H) then searching solutions of (NH) of the form λ(t)X1 (t) +
µ(t)X2 (t); iii) computing etA , and applying Duhamel’s formula.

2
  
2 1 3
Exercise 49 (2015-III-6) (exponential of a 3-3 matrix) Compute etA when A =  0 2 −1 
0 0 2
(hint: A − 2I is nilpotent). Then solve the initial value problem: x0 = 2x + y + 3z, y 0 = 2y − z, z 0 =
2z, x(0) = 1, y(0) = 2, z(0) = 1.

Exercise 50 (system of the form M X 0 (t) = AX(t) + B(t) with M invertible) Solve the system:
 0
2x1 (t) + x02 (t) − 3x1 (t) − x2 (t) = t
x01 (t) + x02 (t) − 4x1 (t) − x2 (t) = et ,

Exercise 51 (nth order equation with polynomial right-hand-side). Consider the nth order equation
an x(n) (t) + ... + a1 x0 (t) + a0 x(t) = Q(t) where Q is a polynomial and a0 ,...an are real numbers
such that an 6= 0. Show that if a0 6= 0, there is a unique polynomial solution and it is of degree
deg(Q). What if a0 = 0 and a1 6= 0? Application: find solutions of x00 (t) + x0 (t) + x(t) = t and
000 00
x (t) + x (t) + x0 (t) = t, then solve these equations.

Exercise 52 (2015-III-10) (resonance) Watch movies on resonance on the web, e.g., at

http://lewebpedagogique.com/physique/quelques-videos-de-resonnances/ .
Let ω > 0 and ν ≥ 0. Solve as a function of ν the differential equation

x00 (t) + ω 2 x(t) = cos(νt).

What does this equation represent (note: students know little physics)? What happens when ν → ω?

Exercise 53 (2015-V-1) (phase portraits and stability) Sketch the phase portrait of the system
X 0 (t) = AX and say if the origin is stable, asymptotically stable, or unstable for:
         
1 0 −1 0 1 0 0 0 0 −1
1) A = ; 2) A = ; 3) A = ; 4) A = ; 5) A =
0 2 0 −2 0 −2 0 −1 0 0

Exercise 54 (damped oscillator) Find the solutions of x00 (t) + αx0 (t) + ω 2 x(t) = 0 with α > 0. Note:
this models a damped oscillator, that is, an oscillator with friction.

Exercise 55 (2015-V-2) (trace-determinant plane and stability) Let A ∈ M2 (R). What can be
said of the stability of the origin in the following cases? 1) detA < 0 ; 2) T r(A) > 0 ; 3) det(A) > 0
and T r(A) < 0 ; 4) detA = 0 et T r(A) < 0.