91Ó°ÊÓ

In Exercises 7-20, solve the given problem. Assume that the functions in each problem have Fourier transforms. Take \(-\infty<0\). $$ t^{2} \frac{\partial u}{\partial x}-\frac{\partial u}{\partial t}=0, \quad u(x, 0)=f(x) . $$

In Exercises 7-12, find the Fourier sine transform of \(f(x)(x>0)\) and write \(f(x)\) as an inverse sine transform. Use a known Fourier transform and (10) when possible. $$ f(x)=e^{-2 x} $$

Find the Fourier integral representation of the given function. $$ f(x)= \begin{cases}x & \text { if }-1

In Exercises 7-12, find the Fourier sine transform of \(f(x)(x>0)\) and write \(f(x)\) as an inverse sine transform. Use a known Fourier transform and (10) when possible. $$ f(x)=\pi e^{-x} $$

In Exercises 7-20, solve the given problem. Assume that the functions in each problem have Fourier transforms. Take \(-\infty<0\). $$ a(t) \frac{\partial u}{\partial x}+\frac{\partial u}{\partial t}=0, \quad u(x, 0)=f(x) $$

Find the Fourier integral representation of the given function. $$ f(x)= \begin{cases}\sin x & \text { if } 0

In Exercises 7-12, find the Fourier sine transform of \(f(x)(x>0)\) and write \(f(x)\) as an inverse sine transform. Use a known Fourier transform and (10) when possible. $$ f(x)= \begin{cases}\sin 2 x & \text { if } 0

In Exercises 7-20, solve the given problem. Assume that the functions in each problem have Fourier transforms. Take \(-\infty<0\). $$ \frac{\partial u}{\partial x}=\frac{\partial u}{\partial t}, \quad \mathrm{u}(x, 0)=f(x) $$

Find the Fourier integral representation of the given function. $$ f(x)= \begin{cases}e^{-k} & \text { if } x>1 \\ 0 & \text { otherwise }\end{cases} $$

In Exercises 7-20, solve the given problem. Assume that the functions in each problem have Fourier transforms. Take \(-\infty<0\). $$ \frac{\partial u}{\partial t}+\sin t \frac{\partial u}{\partial x}=0, \quad u(x, 0)=\sin x $$