Second Order Equations

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 $y'' = -y$.)