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 \(\langle 21 \rangle = \langle m \rangle \).