Problem 5

The Laplace transform $F ( s ) = L [ f ( t ) ]$ of a function $f ( t )$, where $t \geq 0$, is defined as
$$F ( s ) = \int _ { 0 } ^ { \infty } f ( t ) e ^ { - s t } d t$$
Here, $s$ is a complex number, and $e$ is the base of the natural logarithm. Answer the following questions. Show the derivation process with your answer. I. Prove the following relations:

1. $L \left[ t ^ { n } \right] = \frac { n ! } { s ^ { n + 1 } }$, where $n$ is a natural number.
2. $L \left[ \frac { d f ( t ) } { d t } \right] = s F ( s ) - f ( 0 )$, where $f ( t )$ is a differentiable function.
3. $L \left[ e ^ { a t } f ( t ) \right] = F ( s - a )$, where $a$ is a real number.

II. Solve the following differential equation using a Laplace transformation for $t \geq 0$:
$$t \frac { d ^ { 2 } f ( t ) } { d t ^ { 2 } } + ( 1 + 3 t ) \frac { d f ( t ) } { d t } + 3 f ( t ) = 0 , \quad f ( 0 ) = 1 , \left. \quad \frac { d f } { d t } \right| _ { t = 0 } = - 3$$
You can use the relation $L [ t f ( t ) ] = - \frac { d } { d s } F ( s )$, if necessary. III. The point $\mathrm { P } ( x ( t ) , y ( t ) )$, which satisfies the following simultaneous differential equations, passes through the point $( a , b )$ when $t = 0$. $a$ and $b$ are real numbers.
$$\left\{ \begin{array} { l } \frac { d x ( t ) } { d t } = - x ( t ) \\ \frac { d y ( t ) } { d t } = x ( t ) - 2 y ( t ) \end{array} \right.$$

1. Solve the simultaneous differential equations using a Laplace transformation for $t \geq 0$.
2. Express the relation between $x$ and $y$ by eliminating $t$ from the solution of III. 1.
3. For both $( a , b ) = ( 1,1 )$ and $( - 1,1 )$, draw the trajectories of point P when $t$ varies continuously from 0 to infinity.

Problem 5
Consider the continuously differentiable function $f ( x )$ of the real variable $x$. Let $f ( x ) \rightarrow 0$ as $| x | \rightarrow \infty$. $f ( x )$, its derivative $f ^ { \prime } ( x )$, and $x f ( x )$ are absolutely integrable. The Fourier transform of the function $f ( x )$ is denoted by $\mathcal { F } \{ f ( x ) \} ( u )$ or equivalently by $\hat { f } ( u )$, and defined by $$\mathcal { F } \{ f ( x ) \} ( u ) = \hat { f } ( u ) \equiv \frac { 1 } { \sqrt { 2 \pi } } \int _ { - \infty } ^ { \infty } f ( x ) \exp ( - i u x ) \, d x \tag{1}$$ where $u$ is a real variable and $i$ is the imaginary unit. The Fourier transform is defined in the same way for other functions.
I. Express $\mathcal { F } \left\{ f ^ { \prime } ( x ) \right\} ( u )$ in terms of $\hat { f } ( u )$ and $u$.
II. Express $\frac { d \hat { f } ( u ) } { d u }$ in terms of $\mathcal { F } \{ x f ( x ) \} ( u )$.
III. Let the function $f ( x ) = \exp \left( - a x ^ { 2 } \right)$, where $a$ is a positive real constant $( a > 0 )$. The following relation holds for $f ( x )$: $$f ^ { \prime } ( x ) = - 2 a x f ( x ) \tag{2}$$ Apply the Fourier transform on both sides of Eq. (2) to obtain a first-order ordinary differential equation in $\hat { f } ( u )$. Solve this ordinary differential equation to obtain $\hat { f } ( u )$. Note that the integration constant in the solution of this ordinary differential equation can be obtained by calculating $\hat { f } ( 0 )$ with the help of Eq. (1) and the value of the following improper integral: $$\int _ { - \infty } ^ { \infty } \exp \left( - a x ^ { 2 } \right) d x = \sqrt { \frac { \pi } { a } } \tag{3}$$
IV. Consider the function $h ( x , t )$ of the real variables $x$ and $t$. Let $h ( x , t )$ be defined for $- \infty < x < \infty$ and $t \geq 0$, and satisfy the following partial differential equation: $$\frac { \partial h ( x , t ) } { \partial t } = \frac { \partial ^ { 2 } h ( x , t ) } { \partial x ^ { 2 } } \quad ( t > 0 ) \tag{4}$$ given the initial condition $$h ( x , 0 ) = \exp \left( - a x ^ { 2 } \right) \quad ( a > 0 ) \tag{5}$$

1. Apply the Fourier transform with respect to the variable $x$ on both sides of the partial differential equation (4) to obtain an ordinary differential equation with $\hat { h } ( u , t ) \equiv \frac { 1 } { \sqrt { 2 \pi } } \int _ { - \infty } ^ { \infty } h ( x , t ) \exp ( - i u x ) \, d x$ and the independent variable $t$.
2. By solving the ordinary differential equation found in Question IV.1, obtain $\hat { h } ( u , t )$.
3. Use the inverse Fourier transform with respect to the variable $u$ to obtain a solution $h ( x , t )$ satisfying Eq. (4) and Eq. (5).

V. Consider the continuous function $g ( x )$ and its Fourier transform $\hat { g } ( u )$. Let $g ( x ) \rightarrow 0$ as $| x | \rightarrow \infty$ and $g ( x )$ be absolutely integrable. The convolution of the functions $f ( x )$ and $g ( x )$ is defined by $$( f * g ) ( x ) \equiv \int _ { - \infty } ^ { \infty } f ( y ) g ( x - y ) \, d y \tag{6}$$

1. Express $\mathcal { F } \{ ( f * g ) ( x ) \} ( u )$ in terms of $\hat { f } ( u )$ and $\hat { g } ( u )$.
2. Here, the function $h ( x , t )$ satisfies Eq. (4), given the initial condition $h ( x , 0 ) = g ( x )$. Use the result of Question V.1 to find an integral representation of a solution $h ( x , t )$, where $t > 0$.

Consider a real-valued function $f(t)$ for a real variable $t$ defined for $0 \leq t < \infty$. The Laplace transform is defined as
$$\mathcal{L}[f(t)] = F(s) = \int_{0}^{\infty} e^{-st} f(t) \, \mathrm{d}t \tag{1}$$
where $s$ is a complex variable whose real part is positive. Under the condition that the improper integral in the right-hand side does not diverge, answer the following questions.

1. When the conditions $$\lim_{t \rightarrow \infty} e^{-st} f(t) = 0 \quad \text{and} \quad \lim_{t \rightarrow \infty} e^{-st} f^{\prime}(t) = 0$$ are satisfied, show the following equation holds: $$\mathcal{L}\left[f^{\prime\prime}(t)\right] = -f^{\prime}(0) - s f(0) + s^{2} \mathcal{L}[f(t)]$$ Note that $f^{\prime}(t)$ and $f^{\prime\prime}(t)$ are defined as $$f^{\prime}(t) = \frac{\mathrm{d}f(t)}{\mathrm{d}t} \quad \text{and} \quad f^{\prime\prime}(t) = \frac{\mathrm{d}^{2}f(t)}{\mathrm{d}t^{2}}$$
   
2. Calculate the Laplace transform of $g(t) = e^{-at}\cos(\omega t)$ and $h(t) = e^{-at}\sin(\omega t)$ defined for $0 \leq t < \infty$ by showing derivation processes using Equation (1). Note that $a$ and $\omega$ are positive real numbers.
   
3. Solve the differential equation $$f^{\prime\prime}(t) + 6f^{\prime}(t) + 13f(t) = 0$$ where the initial values are $f(0) = 5$ and $f^{\prime}(0) = -11$.