Let the function $c:I \rightarrow \mathbb{R}$ is continuous. Suppose that $u_1$ and $u_2$ are the solutions of $u^\prime +c(t)u = 0$, with $u_1(t_1)=a$ and $u_2(t_2) =b$, where $a$ and $b$ are constants and $t,t_1,t_2$ are members of an interval $I$. Then show that $\mid u_1(t) - u_2(t) \mid \rightarrow 0$ as $\mid t_1 - t_2 \mid \rightarrow 0$ and $\mid a - b\mid \rightarrow 0$ for all $t \in I$.

Consider a linear equation

$u^\prime + a(t)u =0$  on $[0, \infty)$ ,

where $a$ is a complex-valued function with the real part $a_1$ and imaginary part $a_2$. Further, assume that $\int _0 ^\infty \mid a_1(s) \mid ds$ is bounded. Then, show that " for every $\varepsilon >0$, there exists a $\delta = \delta(\varepsilon)>0$ such that $\mid u(t;0,u_0) \mid < \varepsilon$ for all $t \ge 0$, whenever $\mid u_0 \mid = \mid u(0) \mid < \delta(\varepsilon)$ ", where $u(t;0,u_0)$ is the solution passing through $(0,u_0)$. This  property of linear differential equations is usually called the “stability of the null solution”. Show that the null solutions of the following equations are stable:

(i) $x^\prime = -2x;$

(ii) $3x^\prime = -5x ;$

(iii) Give an example of a linear differential equation whose null solution is not stable.

(a) Consider a linear system 

 $u^\prime + c(t)u=0$,

where $c$ is a periodic (complex-valued) function, that is, continuous on $-\infty < t < \infty$. Show that if any nontrivial solution, then

$u(t+\gamma ) = \exp \left [-\int_0^\gamma c(s)ds \right]u(t) ,\gamma >0$,

for all $t$, where $\gamma$ is the period of $c$.

(b) Under the above assumptions, show that any nontrivial solution $u$ of $u^\prime + c(t)u=0$, is periodic with period $\gamma$ if $\int_0^\gamma c(s)ds=0$.

(c) Assuming that $c(t)$ is a constant $c$, what conditions does $c$ have to satisfy in order that any nontrivial solution of $u^\prime +cu =0$ is periodic,

(i) with period $\gamma$;

(ii) with period $n\gamma$, where $n$ is a positive integer.

For the equation 

 $u^\prime +u =d(t)$,

show that there exists a unique bounded solution on $-\infty <t< \infty$, when $\mid d(t) \mid \le M, t \in (-\infty , \infty )$ . If in addition $d$ is periodic then, show that this solution is also periodic.

Consider a non-homogeneous linear differential equation

$u^\prime +c(t)u=d(t)$,

where $c$ and $d$ are given continuous periodic functions on $-\infty<t< \infty$ of period $\gamma$ and $d(t) \neq 0$.

(a) Show that a solution $u$ is periodic of period $\gamma$ if and only if $u(0)=u(\gamma)$.

(b) If there is no nontrivial periodic solution of period $\gamma$ of the corresponding homogeneous equation, then prove that a unique solution of period $\gamma$ for

$u^\prime +c(t)u=d(t)$

exists. Construct an example to illustrate (ii).

Input-output relations: Consider a linear equation

$u^\prime +c(t)u=d(t).$

In many engineering and experimental sciences, the function $d$ is called an “input” and the solution $u$ is termed as an “output” corresponding to the input $d$. In the following cases, determine the output $u$, which is zero at $t=0$:

(a) $u^\prime +3u =5$;

(b) $u^\prime+2u =e^{-t}$ ;

(c) $u^\prime +2u =f(t) , \ where \ f(t)=\left\{ \begin{array}{rcl} 1,\ for \ t<0, \\ 2, \ for \ t \ge 0. \end{array}\right.$

Let $G$ be a finite group, and suppose that for any two subgroups $H$ and $K$ of $G$ either $H \subseteq K \ or \ K \subseteq H$. Prove that $G$ is a cyclic group of prime power order.

Show that the direct product of two infinite cyclic groups in not cyclic.

Determine the number of subgroups of $(\mathbb{Z_{360}},+)$. Also determine the number of automorphisms of $(\mathbb{Z_{360}},+)$.

Prove that the homomorphic image of a cyclic (abelian) group is cyclic (abelian). Give an example to show that if the homomorphic image of a group is cyclic (abelian), the same may not hold for the group itself.

In $(\mathbb{Z_{24}},+)$ find all $m \in \mathbb{Z_{24}}$ such that $\langle 21 \rangle = \langle m \rangle$.