Math Problems Vault

Problem 3.1.1

Find a curve $C$ in $\mathbb {R}^{2}$ , passing through the point $(3,2)$ , with the following property: Let $L(x_0, y_0)$ be the segment of the tangent line to $C$ at $(x_0, y_0)$ which lies in the first quadrant. Then each point $(x_0, y_0)$ of $C$ is the midpoint of $L(x_0, y_0)$ .

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Problem 3.1.2

Solve the differential equation $g' = 2g$ , $g(0) = a$ , where $a$ is a real constant.

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Problem 3.1.3

Let $n$ be an integer larger than $1$ . Is there a differentiable function on $[0, \infty )$ whose derivative equals its $n^{th}$ power and whose value at the origin is positive?

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Problem 3.1.4

(1)(2)

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(1)

For which real numbers $\alpha > 0$ does the differential equation

\frac{dx}{dt} = x^{\alpha }, \quad x(0) = 0,

have a solution on some interval $[0, b]$ , $b > 0$ ?

(2)

For which values of $\alpha$ are there intervals on which two solutions are defined?

Problem 3.1.5

(1)(2)(3)

Consider the differential equation

\frac{dx}{dt} = x^2 + t^2, \quad x(0) = 1.

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(1)

Prove that for some $b > 0$ , there is a solution defined for $t \in [0, b]$ .

(2)

Find an explicit value of $b$ having the property in Part 1.

(3)

Find a $c > 0$ such that there is no solution on $[0, c]$ .

Problem 3.1.6

Solve the differential equation

\frac{dy}{dx} = x^2 y - 3x^2, \quad y(0) = 1.

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Problem 3.1.7

Prove that every solution $x(t)$ ( $t \geq 0$ ) of the differential equation

\frac{dx}{dt} = x^2 - x^6

with $x(0) > 0$ satisfies $\lim_{t \to \infty } x(t) = 1$ .

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Problem 3.1.8

(1)(2)

Consider the differential equation

x' = \frac{x^3-x}{1+e^x}.

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(1)

Find all its constant solutions.

(2)

Discuss $\lim_{t \to \infty } x(t)$ , where $x(t)$ is the solution such that $x(0) = \frac{1}{2}$ .

Problem 3.1.9

Prove that the initial value problem

\frac{dx}{dt} = 3x + 85 \cos x, \quad x(0) = 77,

has a solution $x(t)$ defined for all $t \in \mathbb {R}$ .

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Problem 3.1.10

(1)(2)

Let $f : \mathbb {R} \to \mathbb {R}$ be a continuous nowhere vanishing function, and consider the differential equation

\frac{dy}{dx} = f(y).

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(1)

For each real number $c$ , show that this equation has a unique, continuously differentiable solution $y = y(x)$ on a neighborhood of $0$ which satisfies the initial condition $y(0) = c$ .

(2)

Deduce the conditions on $f$ under which the solution $y$ exists for all $x \in \mathbb {R}$ , for every initial value $c$ .

Problem 3.1.11

Find all solutions to the differential equation

\frac{dy}{dx} = \sqrt{y}, \quad y(0) = 0.

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Problem 3.1.12

Find all solutions $y : \mathbb {R} \to \mathbb {R}$ to

\frac{dy}{dx} = \sqrt{y(y-2)}, \quad y(0) = 0.

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Problem 3.1.13

Find all real valued $C^1$ solutions $y(x)$ of the differential equation

x\frac{dy}{dx} + y = x \quad (-1 < x < 1).

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Problem 3.1.14

Consider the equation

\frac{dy}{dx} = y - \sin y.

Show that there is an $\varepsilon > 0$ such that if $\left|y_0\right| < \varepsilon$ , then the solution $y = f(x)$ with $f(0) = y_0$ satisfies

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

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Problem 3.1.15

(1)(2)

Consider the differential equation

\frac{dy}{dx} = 3xy + \frac{y}{1+y^2}.

Prove

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(1)

For each $n = 1, 2, \ldots$ , there is a unique solution $y = f_n(x)$ defined for $0 \leq x \leq 1$ such that $f_n(0) = 1/n$ .

(2)

$\lim_{n \to \infty } f_n(1) = 0$ .

Problem 3.1.16

Let $y(t)$ be a real valued solution, defined for $0 < t < \infty$ , of the differential equation

\frac{dy}{dt} = e^{-y} - e^{-3y} + e^{-5y}.

Show that $y(t) \to +\infty$ as $t \to +\infty$ .

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Problem 3.1.17

Consider the differential-delay equation given by $y'(t) = -y(t-t_0)$ . Here, the independent variable $t$ is a real variable, the function $y$ is allowed to be complex valued, and $t_0$ is a positive constant. Prove that if $0 < t_0 < \pi /2$ then every solution of the form $y(t) = e^{\lambda t}$ , with $\lambda$ complex, tends to $0$ as $t \to +\infty$ .

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Problem 3.1.18

Prove the following theorem, or find a counterexample: If $p$ and $q$ are continuous real valued functions on $\mathbb {R}$ such that $\left|q(x)\right| \leq \left|p(x)\right|$ for all $x$ , and if every solution $f$ of the differential equation

f' + qf = 0

satisfies $\lim_{x \to +\infty } f(x) = 0$ , then every solution $f$ of the differential equation

f' + pf = 0

satisfies $\lim_{x \to +\infty } f(x) = 0$ .

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Problem 3.1.19

Discuss the solvability of the differential equation

(e^x \sin y)(y')^3 + (e^x \cos y)y' + e^y \tan x = 0

with the initial condition $y(0) = 0$ . Does a solution exist in some interval about $0$ ? If so, is it unique?

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Problem 3.1.20

Let $f$ and $g$ be positive continuous functions on $\mathbb {R}$ , with $g \leq f$ everywhere. Assume the initial value problem

\frac{dx}{dt} = f(x), \quad x(0) = 0,

has a solution defined on all of $\mathbb {R}$ . Prove that the initial value problem

\frac{dx}{dt} = g(x), \quad x(0) = 0,

also has a solution defined on all of $\mathbb {R}$ .

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Problem 3.1.21

Let $f : \mathbb {R} \to \mathbb {R}$ be a bounded continuously differentiable function. Show that every solution of $y'(x) = f(y(x))$ is monotone.

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