Find all possible values of \(\int_\gamma \frac{dz}{1+z^2}\) where \(\gamma\) is any closed rectifiable curve in \(\mathbb{C}\) not passing through \(\pm i\).

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.