Differential Equations

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

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?

Consider the differential equation

Solve the differential equation

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

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

Consider the differential equation

Prove that the initial value problem

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

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

Find all solutions to the differential equation

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

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

Consider the equation

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

Consider the differential equation

Prove

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

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

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 $t \to +\infty$.

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

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

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

Discuss the solvability of the differential equation

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

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

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

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

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.

Suppose that $f''(x) = (x^2-1)f(x)$ for all $x \in \mathbb {R}$, and that $f(0) = 1$, $f'(0) = 0$. Show that $f(x) \to 0$ as $x \to \infty$.

Find the solution of the differential equation

subject to the conditions

Find all solutions of the differential equation

subject to the condition $x(0) = 1$ and $x'(0) = 0$.

Let $x : \mathbb {R} \to \mathbb {R}$ be a solution to the differential equation

Prove that the function $f : \mathbb {R} \to \mathbb {R}$,

attains a maximum value.

Let $x(t)$ be the solution of the differential equation

with initial conditions $x(0) = 0$ and $x'(0) = 0$. Show that for suitable constants $\alpha$ and $\delta$,

Let $y = y(x)$ be a solution of the differential equation $y'' = -\left|y\right|$ with $-\infty < x < \infty$, $y(0) = 1$ and $y'(0) = 0$.

Determine all real numbers $L > 1$ so that the boundary value problem

has a nonzero solution.

For which real values of $p$ does the differential equation

admit solutions $y = f(x)$ with infinitely many critical points?

Let $p$, $q$ and $r$ be continuous real valued functions on $\mathbb {R}$, with $p > 0$. Prove that the differential equation

is equivalent to (i.e., has exactly the same solutions as) a differential equation of the form

where $a$ is continuously differentiable and $b$ is continuous.

Let the function $x(t)$ ($-\infty < t < \infty$) be a solution of the differential equation

such that $x(0) = x(1) = 0$. (Here, $b$ and $c$ are real constants.) Prove that $x(n) = 0$ for every integer $n$.

Let $k$ be a positive integer. For which values of the real number $c$ does the differential equation

have a solution satisfying $x(0) = x(2\pi k) = 0$?

Consider the second-order linear differential equation

Here, the independent variable $t$ varies over $\mathbb {R}$, the unknown function $x$ is assumed to be real valued, and $p$ and $q$ are continuous functions on $\mathbb {R}$. Assume that the solutions of $(*)$ are defined for all $t$ (which is actually guaranteed by the theory), and that the solution set is translation invariant: if $f$ is a solution and $s$ is a real number, then the function $g(t) = f(t+s)$ is also a solution. Prove that $p$ and $q$ are constant.

Let $h > 0$ be given. Consider the linear difference equation

(Note the analogy with the differential equation $y'' = -y$.)

Let $E$ be the set of functions $f : \mathbb {R} \to \mathbb {R}$ which are solutions to the differential equation $f''' + f'' - 2f = 0$.

Find a function $y(x)$ such that $y^{(4)} + y = 0$ for $x \geq 0$, $y(0) = 0$, $y'(0) = 1$ and $\lim_{x \to \infty } y(x) = \lim_{x \to \infty } y'(x) = 0$.

Let $V$ be a finite-dimensional linear subspace of $C^{\infty }(\mathbb {R})$ (the space of complex valued, infinitely differentiable functions). Assume that $V$ is closed under $D$, the operator of differentiation (i.e., $f \in V \Rightarrow Df \in V$). Prove that there is a constant coefficient differential operator

such that $V$ consists of all solutions of the differential equation $Lf = 0$.

Let $y : \mathbb {R} \to \mathbb {R}$ be a three times differentiable function satisfying the differential equation $y''' - y = 0$. Suppose that $\lim_{x \to \infty } y(x) = 0$. Find real numbers $a, b, c$, and $d$, not all zero, such that $ay(0) + y'(0) + cy''(0) = d$.

Consider the system of differential equations:

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

and also

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

Consider the differential equations

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.

Consider the system of differential equations

Show that the system of differential equations

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

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

where

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

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

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 $t=2$.

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

Consider the differential equation

Consider an autonomous system of differential equations

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

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

where

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

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

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 $a$).

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

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

Prove that for each $k$,

Consider the vector differential equation

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 $t > 0$.

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

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

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

Prove that

Let

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