Math Problems Vault

Problem 3.4.1

Consider the system of differential equations:

\frac{dx}{dt} = y + tz

\frac{dy}{dt} = z + t^2 x

\frac{dz}{dt} = x + e^t y.

Prove there exists a solution defined for all $t \in [0,1]$ , such that

\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} \begin{pmatrix} x(0) \\ y(0) \\ z(0) \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}

and also

\int _0^1 \left(x(t)^2 + y(t)^2 + z(t)^2\right) \, dt = 1.

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Problem 3.4.2

Find all pairs of $C^{\infty }$ functions $x(t)$ and $y(t)$ on $\mathbb {R}$ satisfying

x'(t) = 2x(t) - y(t), \quad y'(t) = x(t).

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Problem 3.4.3

Consider the differential equations

\frac{dx}{dt} = -x+y, \quad \frac{dy}{dt} = \log (20+x) - y.

Let $x(t)$ and $y(t)$ be a solution defined for all $t \geq 0$ with $x(0) > 0$ and $y(0) > 0$ . Prove that $x(t)$ and $y(t)$ are bounded.

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Problem 3.4.4

(1)(2)

Consider the system of differential equations

\frac{dx}{dt} = y + x(1-x^2-y^2)

\frac{dy}{dt} = -x + y(1-x^2-y^2).

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(1)

Show that for any $x_0$ and $y_0$ , there is a unique solution $(x(t), y(t))$ defined for all $t \in \mathbb {R}$ such that $x(0) = x_0$ , $y(0) = y_0$ .

(2)

Show that if $x_0 \neq 0$ and $y_0 \neq 0$ , the solution referred to in Part 1 approaches the circle $x^2+y^2=1$ as $t \to \infty$ .

Problem 3.4.5

Show that the system of differential equations

\frac{d}{dt}\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 0 & 1 & 0 \\ 2 & 0 & 0 \\ 0 & 0 & 3 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix}

has a solution which tends to $\infty$ as $t \to -\infty$ and tends to the origin as $t \to +\infty$ .

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Problem 3.4.6

Let $x(t)$ be a nontrivial solution to the system

\frac{dx}{dt} = Ax,

where

A = \begin{pmatrix} 1 & 6 & 1 \\ -4 & 4 & 11 \\ -3 & -9 & 8 \end{pmatrix}.

Prove that $\left\| x(t)\right\|$ is an increasing function of $t$ . (Here, $\left\| \cdot \right\|$ denotes the Euclidean norm.)

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Problem 3.4.7

Consider the solution curve $(x(t), y(t))$ to the equations

\frac{dx}{dt} = 1 + \frac{1}{2}x^2 \sin y

\frac{dy}{dt} = 3 - x^2

with initial conditions $x(0) = 0$ and $y(0) = 0$ . Prove that the solution must cross the line $x=1$ in the $xy$ plane by the time $t=2$ .

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Problem 3.4.8

(1)(2)

Consider the differential equation $x'' + x' + x^3 = 0$ and the function $f(x,x') = (x+x')^2 + (x')^2 + x^4$ .

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(1)

Show that $f$ decreases along trajectories of the differential equation.

(2)

Show that if $x(t)$ is any solution, then $(x(t), x'(t))$ tends to $(0,0)$ as $t \to \infty$ .

Problem 3.4.9

(1)(2)

Consider the differential equation

\frac{dx}{dt} = y, \quad \frac{dy}{dt} = -ay - x^3 - x^5, \quad \text{where } a > 0.

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(1)

Show that

F(x,y) = \frac{y^2}{2} + \frac{x^4}{4} + \frac{x^6}{6}

decreases along solutions.

(2)

Show that for any $\varepsilon > 0$ , there is a $\delta > 0$ such that whenever $\left\| (x(0),y(0))\right\| < \delta$ , there is a unique solution $(x(t),y(t))$ of the given equations with the initial condition $(x(0),y(0))$ which is defined for all $t \geq 0$ and satisfies $\left\| (x(t),y(t))\right\| < \varepsilon$ .

Problem 3.4.10

(1)(2)

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(1)

Let $u(t)$ be a real valued differentiable function of a real variable $t$ which satisfies an inequality of the form

u'(t) \leq au(t), \quad t \geq 0, \quad u(0) \leq b,

where $a$ and $b$ are positive constants. Starting from first principles, derive an upper bound for $u(t)$ for $t > 0$ .

(2)

Let $x(t) = (x_1(t), x_2(t), \ldots , x_n(t))$ be a differentiable function from $\mathbb {R}$ to $\mathbb {R}^{n}$ which satisfies a differential equation of the form

x'(t) = f(x(t)),

where $f : \mathbb {R}^{n} \to \mathbb {R}^{n}$ is a continuous function. Assuming that $f$ satisfies the condition

\left\langle f(y), y \right\rangle \leq \left\| y\right\| ^2, \quad y \in \mathbb {R}^{n}

derive an inequality showing that the norm $\left\| x(t)\right\|$ grows, at most, exponentially.

Problem 3.4.11

(1)(2)

Consider an autonomous system of differential equations

\frac{dx_i}{dt} = F_i(x_1, \ldots , x_n),

where $F = (F_1, \ldots , F_n) : \mathbb {R}^{n} \to \mathbb {R}^{n}$ is a $C^1$ vector field.

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(1)

Let $U$ and $V$ be two solutions on $a < t < b$ . Assuming that

\left\langle DF(x)z, z \right\rangle \leq 0

for all $x, z$ in $\mathbb {R}^{n}$ , show that $\left\| U(t)-V(t)\right\|^2$ is a decreasing function of $t$ .

(2)

Let $W(t)$ be a solution defined for $t > 0$ . Assuming that

\left\langle DF(x)z, z \right\rangle \leq -\left\| z\right\| ^2,

show that there exists $C \in \mathbb {R}^{n}$ such that

\lim _{t \to \infty } W(t) = C.

Problem 3.4.12

(1)(2)

Let $V : \mathbb {R}^{n} \to \mathbb {R}$ be a $C^1$ function and consider the system of second order differential equations

x_i''(t) = f_i(x(t)), \quad 1 \leq i \leq n,

where

f_i = -\frac{\partial V}{\partial x_i}.

Let $x(t) = (x_1(t), \ldots , x_n(t))$ be a solution of this system on a finite interval $a < t < b$ .

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(1)

Show that the function

H(t) = \frac{1}{2} \left\langle x'(t), x'(t) \right\rangle + V(x(t))

is constant for $a < t < b$ .

(2)

Assuming that $V(x) \geq M > -\infty$ for all $x \in \mathbb {R}^{n}$ , show that $x(t)$ , $x'(t)$ , and $x''(t)$ are bounded on $a < t < b$ , and then prove all three limits

\lim _{t \to b} x(t), \quad \lim _{t \to b} x'(t), \quad \lim _{t \to b} x''(t)

exist.

Problem 3.4.13

For $\lambda$ a real number, find all solutions of the integral equations

\varphi (x) = e^x + \lambda \int _0^x e^{(x-y)}\varphi (y) \, dy, \quad 0 \leq x \leq 1,

\psi (x) = e^x + \lambda \int _0^1 e^{(x-y)}\psi (y) \, dy, \quad 0 \leq x \leq 1.

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Problem 3.4.14

Let $V$ be a finite-dimensional vector space (over $\mathbb {C}$ ) of $C^{\infty }$ complex valued functions on $\mathbb {R}$ (the linear operations being defined pointwise). Prove that if $V$ is closed under differentiation (i.e., $f'(x)$ belongs to $V$ whenever $f(x)$ does), then $V$ is closed under translations (i.e., $f(x+a)$ belongs to $V$ whenever $f(x)$ does, for all real numbers $a$ ).

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Problem 3.4.15

Describe all three dimensional vector spaces $V$ of $C^{\infty }$ complex valued functions on $\mathbb {R}$ that are invariant under the operator of differentiation.

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Problem 3.4.16

Let the real valued functions $f_1, \ldots , f_{n+1}$ on $\mathbb {R}$ satisfy the system of differential equations

f_{k+1}' + f_k' = (k+1)f_{k+1} - kf_k, \quad k = 1, \ldots , n

f_{n+1}' = -(n+1)f_{n+1}.

Prove that for each $k$ ,

\lim _{t \to \infty } f_k(t) = 0.

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Problem 3.4.17

Consider the vector differential equation

\frac{dx(t)}{dt} = A(t)x(t)

where $A$ is a smooth $n \times n$ function on $\mathbb {R}$ . Assume $A$ has the property that $\left\langle A(t)y, y \right\rangle \leq c\left\| y\right\|^2$ for all $y$ in $\mathbb {R}^{n}$ and all $t$ , where $c$ is a fixed real number. Prove that any solution $x(t)$ of the equation satisfies $\left\| x(t)\right\| \leq e^{ct}\left\| x(0)\right\|$ for all $t > 0$ .

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Problem 3.4.18

(1)(2)(3)

Let $W$ be a real $3 \times 3$ antisymmetric matrix, i.e., $W^t = -W$ . Let the function

X(t) = \begin{pmatrix} x_1(t) \\ x_2(t) \\ x_3(t) \end{pmatrix}

be a real solution of the vector differential equation $\frac{dX}{dt} = WX$ .

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(1)

Prove that $\left\| X(t)\right\|$ , the Euclidean norm of $X(t)$ , is independent of $t$ .

(2)

Prove that if $v$ is a vector in the null space of $W$ , then $X(t) \cdot v$ is independent of $t$ .

(3)

Prove that the values $X(t)$ all lie on a fixed circle in $\mathbb {R}^{3}$ .

Problem 3.4.19

For each $t \in \mathbb {R}$ , let $P(t)$ be a symmetric real $n \times n$ matrix whose entries are continuous functions of $t$ . Suppose for all $t$ that the eigenvalues of $P(t)$ are all $\leq -1$ . Let $x(t) = (x_1(t), \ldots , x_n(t))$ be a solution of the vector differential equation

\frac{dx}{dt} = P(t)x.

Prove that

\lim _{t \to \infty } x(t) = 0.

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Problem 3.4.20

Let

A = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{pmatrix}, \qquad B = \begin{pmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{pmatrix}

Find the general solution of the matrix differential equation $\frac{dX}{dt} = AXB$ for the unknown $4 \times 4$ matrix function $X(t)$ .

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