Chapter 23

Linear Differential Equations

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Problem 23.1.1

Solve the initial value problem of $d\overrightarrow {x}/dt = A\overrightarrow {x}$ , where

A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} .

Problem 23.1.2

(i)(ii)

(i)

Solve the initial value problem of $d\overrightarrow {x}/dt = A\overrightarrow {x}$ , where

A = \begin{pmatrix} a & c \\ 0 & b \end{pmatrix}, \quad a, b, c \in \mathbb {R} .

(ii)

Let

A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} .

Calculate $\exp (tA)$ , where $t \in \mathbb {R}$ . Find the solution of the initial value problem of the differential equation

\begin{pmatrix} du_{1}/dt \\ du_{2}/dt \end{pmatrix} = A \begin{pmatrix} u_{1} \\ u_{2} \end{pmatrix}

with the initial conditions $u_{1}(t=0) = u_{10}$ , $u_{2}(t=0) = u_{20}$ . Use the result from (i).

Problem 23.1.3

Show that the $n$ -th order differential equation

\frac{d^{n}x}{dt^{n}} = c_{0}x + c_{1}\frac{dx}{dt} + \cdots + c_{n-1}\frac{d^{n-1}x}{dt^{n-1}}, \quad c_{j} \in \mathbb {R}

can be written as a system of first-order differential equation.

Problem 23.1.4

Let $A$ , $X$ , $F$ be $n \times n$ matrices. Assume that the matrix elements of $X$ and $F$ are differentiable functions of $t$ . Consider the initial-value linear matrix differential equation with an inhomogeneous part

\frac{dX(t)}{dt} = AX(t) + F(t), \quad X(t_{0}) = C .

Find the solution of this matrix differential equation.

Problem 23.1.5

Let $A$ , $B$ , $C$ , $Y$ be $n \times n$ matrices. We know that $AY + YB = C$ can be written as

((I_{n} \otimes A) + (B^{T} \otimes I_{n}))\operatorname {vec}(Y) = \operatorname {vec}(C)

where $\otimes$ denotes the Kronecker product. The vec operation is defined as

\operatorname {vec}(Y) := (y_{11}, \ldots , y_{n1}, y_{12}, \ldots , y_{n2}, \ldots , y_{1n}, \ldots , y_{nn})^{T} .

Apply the vec operation to the matrix differential equation

\frac{d}{dt}X(t) = AX(t) + X(t)B

where $A$ , $B$ are $n \times n$ matrices and the initial matrix $X(t=0) \equiv X(0)$ is given. Find the solution of this differential equation.

Problem 23.1.6

The motion of a charge $q$ in an electromagnetic field is given by

m\frac{d\overrightarrow {v}}{dt} = q(\overrightarrow {E} + \overrightarrow {v} \times \overrightarrow {B})

where $m$ denotes the mass and $\overrightarrow {v}$ the velocity. Assume that

\overrightarrow {E} = \begin{pmatrix} E_{1} \\ E_{2} \\ E_{3} \end{pmatrix}, \qquad \overrightarrow {B} = \begin{pmatrix} B_{1} \\ B_{2} \\ B_{3} \end{pmatrix}

are constant fields. Find the solution of the initial value problem.

Problem 23.1.7

Consider a system of linear ordinary differential equations with periodic coefficients

\frac{d\overrightarrow {u}}{dt} = A(t)\overrightarrow {u}

where $A(t)$ is an $n \times n$ matrix of periodic functions with a period $T$ . From Floquet theory we know that any fundamental $n \times n$ matrix $\Phi (t)$ , which is defined as a nonsingular matrix satisfying the matrix differential equation

\frac{d\Phi (t)}{dt} = A(t)\Phi (t)

can be expressed as

\Phi (t) = P(t)\exp (tR) .

Here $P(t)$ is a nonsingular $n \times n$ matrix of periodic functions with the same period $T$ , and $R$ , a constant matrix, whose eigenvalues are called the characteristic exponents of the periodic system. Let

\overrightarrow {v}(t) = P^{-1}(t)\overrightarrow {u}(t) .

Show that $\overrightarrow {v}$ satisfies the system of linear differential equations with constant coefficients

\frac{d\overrightarrow {v}(t)}{dt} = R\overrightarrow {v}(t) .

Problem 23.1.8

Consider a system of linear ordinary differential equations with periodic coefficients

\frac{d\overrightarrow {u}}{dt} = A(t)\overrightarrow {u}, \qquad \overrightarrow {u} = \begin{pmatrix} u_{2} \\ u_{2} \end{pmatrix}

where $A(t)$ is a $2 \times 2$ matrix of periodic functions with period $T$ . By the classical Floquet theory, any fundamental matrix $\Phi (t)$ , which is defined as a nonsingular matrix satisfying the matrix differential equation

\frac{d\Phi }{dt} = A(t)\Phi (t)

can be expressed as

\Phi (t) = P(t)\exp (TR) .

Here $P(t)$ is nonsingular matrix of periodic functions with the same period $T$ , and $R$ , a constant matrix, whose eigenvalues $\lambda_{1}$ and $\lambda_{2}$ are called the characteristic exponents of the periodic system. For a choice of fundamental matrix $\Phi (t)$ , we have

\exp (TR) = \Phi (t_{0})\Phi (t_{0} + T)

which does not depend on the initial time $t_{0}$ . The matrix $\exp (TR)$ is called the monodromy matrix of the periodic system. Calculate

\operatorname {tr}(\exp (TR)) .

Problem 23.1.9

(i)(ii)

Consider the autonomous system of nonlinear first-order ordinary differential equations

\begin{align} \frac{dx_{1}}{dt} & = a(x_{2} - x_{1}) = f_{1}(x_{1}, x_{2}, x_{3}) \\ \frac{dx_{2}}{dt} & = (c-a)x_{1} + cx_{2} - x_{1}x_{3} = f_{2}(x_{1}, x_{2}, x_{3}) \\ \frac{dx_{3}}{dt} & = -bx_{3} + x_{1}x_{2} = f_{3}(x_{1}, x_{2}, x_{3}) \end{align}

where $a > 0$ , $b > 0$ and $c$ are real constants with $2c > a$ .

(i)

The fixed points are defined as the solutions of the system of equations

\begin{align} f_{1}(x_{1}^{*}, x_{2}^{*}, x_{3}^{*}) & = a(x_{2}^{*} - x_{1}^{*}) = 0 \\ f_{2}(x_{1}^{*}, x_{2}^{*}, x_{3}^{*}) & = (c-a)x_{1}^{*} + cx_{2}^{*} - x_{1}^{*}x_{3}^{*} = 0 \\ f_{3}(x_{1}^{*}, x_{2}^{*}, x_{3}^{*}) & = -bx_{3}^{*} + x_{1}^{*}x_{2}^{*} = 0 . \end{align}

Find the fixed points. Obviously $(0,0,0)$ is a fixed point.

(ii)

The linearized equation (or variational equation) is given by

\begin{pmatrix} dy_{1}/dt \\ dy_{2}/dt \\ dy_{3}/dt \end{pmatrix} = A \begin{pmatrix} y_{1} \\ y_{2} \\ y_{3} \end{pmatrix}

where the $3 \times 3$ matrix $A$ is given by

A_{\overrightarrow {x}=\overrightarrow {x}^{*}} = \begin{pmatrix} \partial f_{1}/\partial x_{1} & \partial f_{1}/\partial x_{2} & \partial f_{1}/\partial x_{3} \\ \partial f_{2}/\partial x_{1} & \partial f_{2}/\partial x_{2} & \partial f_{2}/\partial x_{3} \\ \partial f_{3}/\partial x_{1} & \partial f_{3}/\partial x_{2} & \partial f_{3}/\partial x_{3} \end{pmatrix}_{\overrightarrow {x}=\overrightarrow {x}^{*}}

where $\overrightarrow {x} = \overrightarrow {x}^{*}$ indicates to insert one of the fixed points into $A$ . Calculate $A$ and insert the first fixed point $(0,0,0)$ . Calculate the eigenvalues of $A$ . If all eigenvalues have negative real part then the fixed point is stable. Thus study the stability of the fixed point.

Problem 23.1.10

Let $A$ be an $n \times n$ matrix over $\mathbb {R}$ . Consider the initial value problem of the system of linear differential equations

\frac{d\overrightarrow {u}(t)}{dt} + A\overrightarrow {u}(t) = \overrightarrow {g}(t), \qquad \overrightarrow {u}(0) = \overrightarrow {u}_{0}

where $\overrightarrow {g}(t) = (g_{1}(t), g_{2}(t), \ldots , g_{n}(t))^{T}$ . The solution of the initial value problem is

\overrightarrow {u}(t) = e^{-tA}\overrightarrow {u}_{0} + \int _{0}^{t} e^{-(t-\tau )A}\overrightarrow {g}(\tau )\, d\tau .

Apply it to the $3 \times 3$ matrices

A = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix}

with initial values $\overrightarrow {u}_{0} = (1 \ 1 \ 1)^{T}$ and $\overrightarrow {g}(t) = (1 \ 0 \ 1)^{T}$ .

Discretize the system with the implicit Euler method with step size $h$ and $h = 0.1$ . One has

(I_{n} + hA)\overrightarrow {u}_{j} = \overrightarrow {u}_{j-1} + h\overrightarrow {g}(t_{j}), \qquad j = 1, \ldots , M

with

(A + hI_{3})^{-1} = \frac{1}{h^{2}(h+1) - h - 1} \begin{pmatrix} h(h+1) & 0 & -h-1 \\ 0 & h^{2}-1 & 0 \\ -h-1 & 0 & h(h^{2}+1) \end{pmatrix} .

Compare the two solutions for the matrix $A$ .

Problem 23.1.11

(i)(ii)

Let $L$ and $K$ be two $n \times n$ matrices. Assume that the entries depend on a parameter $t$ and are differentiable with respect to $t$ . Assume that $K^{-1}(t)$ exists for all $t$ . Assume that the time-evolution of $L$ is given by

L(t) = K(t)L(0)K^{-1}(t) .

(i)

Show that $L(t)$ satisfies the matrix differential equation

\frac{dL}{dt} = [L, B](t)

where $[\, , \, ]$ denotes the commutator and

B = -\frac{dK}{dt}K^{-1}(t) .

(ii)

Show that if $L(t)$ is hermitian and $K(t)$ is unitary, then the matrix $B(t)$ is skew-hermitian.

Problem 23.1.12

Solve the initial value problem for the matrix differential equation

[B, A(\epsilon )] = \frac{dA(\epsilon )}{d\epsilon }

where $A(\epsilon )$ and $B = \sigma_{1}$ are $2 \times 2$ matrices.

Problem 23.1.13

Let $a, b \in \mathbb {R}$ . Consider the linear matrix differential equation

\frac{d^{2}X}{dt^{2}} + a\frac{dX}{dt} + bX = 0 .

Find the solution of the initial value problem.

Problem 23.1.14

Let $A$ be an $n \times n$ matrix over $\mathbb {R}$ . The autonomous system of first-order differential equations $d\overrightarrow {u}/dt = A\overrightarrow {u}$ admits the solution of the initial value problem $\overrightarrow {u}(t) = \exp (A)\overrightarrow {u}(0)$ . Differentiation of the differential equations yields the second-order system

\frac{d^{2}\overrightarrow {u}}{dt^{2}} = A\frac{d\overrightarrow {u}}{dt} = A^{2}\overrightarrow {u} .

Thus we can write

\frac{d\overrightarrow {u}}{dt} = \overrightarrow {v} = A\overrightarrow {u}, \qquad \frac{d\overrightarrow {v}}{dt} = A^{2}\overrightarrow {u} = A\overrightarrow {v}

or in matrix form

\begin{pmatrix} d\overrightarrow {u}/dt \\ d\overrightarrow {v}/dt \end{pmatrix} = \begin{pmatrix} 0_{n} & I_{n} \\ A^{2} & 0_{n} \end{pmatrix} \begin{pmatrix} \overrightarrow {u}(0) \\ \overrightarrow {v}(0) \end{pmatrix} .

Find the solution of the initial value problem. Assume that $A$ is invertible.

Problem 23.1.15

(i)(ii)

(i)

Let $\overrightarrow {u} = (u_{1}, \ldots , u_{n})^{T}$ and $A$ be an $n \times n$ matrix over $\mathbb {R}$ . Solve the initial value problem of the system of linear differential equations

\frac{d\overrightarrow {u}}{dt} = A\overrightarrow {u}

using the Laplace transform, where $\overrightarrow {u}(t=0) = \overrightarrow {u}(0)$ . Note that (applying integration by parts)

\mathcal{L}(d\overrightarrow {u}/dt)(s) = \int _{0}^{\infty } e^{-st}\frac{d\overrightarrow {u}}{dt}\, dt = e^{-st}\overrightarrow {u}(t)\Big|_{t=0}^{t=\infty } + s\int _{0}^{\infty } e^{-st}\overrightarrow {u}(t)\, dt = s\overrightarrow {U}(s) - \overrightarrow {u}(0) .

(ii)

Apply it to the $2 \times 2$ Hadamard matrix

H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} .

Section 23.2Supplementary Problems[2/0%]

Problem 23.2.1

Consider the Lotka-Volterra model

\frac{du_{1}}{dt} = f_{1}(u_{1}, u_{2}) = u_{1} - u_{1}u_{2}, \qquad \frac{du_{2}}{dt} = f_{2}(u_{1}, u_{2}) = -u_{2} + u_{1}u_{2}

with the fixed point $(u_{1}^{*} = 1, u_{2}^{*} = 1)$ . The linearized equation (variational equation) with

\frac{\partial f_{1}}{\partial x_{1}} = 1 - u_{2}, \quad \frac{\partial f_{1}}{\partial x_{2}} = -u_{1}, \quad \frac{\partial f_{2}}{\partial x_{1}} = u_{2}, \quad \frac{\partial f_{2}}{\partial x_{2}} = -1 + u_{1}

is given by

\begin{pmatrix} dy_{1}/dt \\ dy_{2}/dt \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} y_{1} \\ y_{2} \end{pmatrix} .

Solve this system of linear differential equations. First show that the eigenvalues of the $2 \times 2$ matrix are given by $\pm i$ .

Problem 23.2.2

Consider the initial problem of the matrix differential equation

\frac{dX}{dt} = A(t)X, \qquad X(0) = I_{n}

where $A(t)$ is an $n \times n$ matrix which depends smoothly on $t$ . It is known that the solution of this matrix differential equation can locally be written as

X(t) = \exp (\Omega (t))

where $\Omega (t)$ is obtained as an infinite series

\Omega (t) = \sum _{k=1}^{\infty } \Omega _{k}(t) .

This is the so-called Magnus expansion. Implement this recursion in SymbolicC++ and apply it to

A(t) = \begin{pmatrix} \cos (\omega t) & -\sin (\omega t) \\ \sin (\omega t) & \cos (\omega t) \end{pmatrix}

where $\omega$ is a constant frequency.