Evaluate $\int_\gamma \frac{e^z -e^{-z}}{z^4}dz$ where $\gamma$ is one of the following curves.

Let $\alpha(t)=e^{it}$ and $\beta(t)=(3/2)+ 3e^{it}$ for $t$ in the interval $[0,2\pi]$. Show that $\alpha$ and $\beta$ are freely homotopic on $A=\mathbb{C}\backslash\{0\}$.

Note: Homotopy means deforming one path to another. Freely homotopic means defining a function involving both the paths, as given in the definition of homotopy.

Let $\alpha:[a,b] \mapsto A$ and $\beta :[a,b] \mapsto B$ are closed paths in a set $A$ that is starlike with respect to a point $z_0$. Show that $\alpha$ and $\beta$ are freely homotopic in $A$ .

Prove that

$u_c(t) = \left\{ \begin{array} 00 \ for \ 0 \le t \le c, \\ (t-3)^3, \ for \ c\le t < \infty , c>0, \end{array}\right.$

 is a solution of  $u^\prime = 3u^{2/3},u(0)=0 .$

(a) Show that all the solutions of the linear equation $u^\prime +a(t)u = 0, \ t \in [0,\infty)$, where $a:[0,\infty) \rightarrow [0,\infty)$ is a continuous function, are always bounded if $\int_0^\infty a(s)ds < \infty$ .

(b) Consider the equation $(1+t)^2u^\prime - u = 0, \ t \in [0,\infty)$ .

(i) Find the unique solution.

(ii) Show that $\mid u(0) \mid \le \mid u(t) \mid < \mid u(0) \mid e$, for all $t \in [0 ,\infty)$ . How you are connecting this result with the problem $(2) (a)$ ?

Let $u$ denotes the solution of the initial value problem:

                 $u^\prime +cu = d(t) , \ u(0)=0$ ,

where $c$ is a complex constant. Let $d$ be a continuous function and bounded by $M$ on

$[0,\infty)$. Then, prove that

                      $\mid u(t) \mid \le \frac{M}{\mid Re(c) \mid }, \ for \ all \ t \in [0,\infty).$

Let $c,d:I \rightarrow \mathbb{R}$ be continuous functions. Let $u_1,u_2$ denote the solutions of the initial value problem

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

with $u_1(t_0)=u_{10}$ and $u_2(t_0)=u_{20}$   on an interval $I$ containing $t_0$. Then, show that

$\mid u_1(t) - u_2(t) \mid= \mid u_{10} -u_{20} \mid \exp \left [ - \int_{t_0}^t c(s)ds \right ] ,t \in I ,$,

and, hence, show that $u_1(t) = u_1(t;t_0 , u_{10} )$ is a continuous function of $x_{10}$ for any fixed $t$ and $t_0$. Thus, conclude that $u(t) = u(t;t_0,u_0)$ of a linear equation is a continuous function of an initial parameter $u_0$.

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.