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

Show that \((\mathbb{Q^+}, .)\), the multiplicative group of positive rational numbers, is not a cyclic group.

Let \(f:G \rightarrow G^\prime \) be a homomorphism of groups. Show that if \(H^\prime \unlhd G^\prime\) , then \(f^{-1}(H^\prime ) \unlhd G\) and \(\ker f \subseteq f^{-1}(H^\prime)\). Further show that if \(f\) is surjective and \(H \unlhd G \), then \(f(H) \unlhd G^\prime \) .

Show by an example that we can find groups \(G,H,K\) such that \(K \unlhd G\) and \(H \unlhd K\), but \(H\) is not a normal subgroup of \(G\).

Give an example of a group \(G\) and a subgroup \(H\) of \(G\) such that for some \(a \in G , aHa^{-1} \subsetneq H\).