91Ó°ÊÓ

Find a general solution of the ODE \(y^{\prime \prime}+a^{2} y=r(t)\) with \(r(f)\) as given. (Show the details of your work.) $$r(t)=\left\\{\begin{array}{rcc}t+\pi & \text { if } & -\pi

Find the Fourier transform of \(f(x)\) (without using Table \(\mathbb{U}\) in Sec. 11.10 ). Show the details. $$f(x)=\left\\{\begin{aligned}-1 & \text { if }-1

Find the Fourier series of the function \(f(x)\), of period \(p=2 L\), and sketch or graph the first three partial sums. (Show the details of your work.) $$f(x)=1-x^{2} \quad(-1 < x < 1), \quad p=2$$

Find the complex Fourier series of the following functions. $$f(x)=x \quad(-\pi

Represent \(f(x)\) as an integral (11). $$f(x)=\left\\{\begin{array}{ccc} x / 2 & \text { if } & 0 < x < 1 \\ 1-x / 2 & \text { if } & 1 < x < 2 \\ 0 & \text { if } & x > 2 \end{array}\right.$$

Is the given function even or odd? Find its Fourier series. Sketch or graph the function and some partial sums. (Show the details of your work.) $$f(x)=x-|x| \quad(-\pi < x < \pi)$$

$$f(x)=\left\\{\begin{aligned} -x^{8} & \text { if }-\pi

Find a general solution of the ODE \(y^{\prime \prime}+a^{2} y=r(t)\) with \(r(f)\) as given. (Show the details of your work.) $$r(t)=\frac{\pi}{4}|\sin t| \text { if }-\pi

Find the Fourier series of the function \(f(x)\), of period \(p=2 L\), and sketch or graph the first three partial sums. (Show the details of your work.) $$\begin{array}{l} f(x)=-x(-1 < x < 0), \quad f(x)=x \quad(0 < x < 1), \\ f(x)=1 \quad(1 < x < 3), \quad p=4 \end{array}$$

Find the Fourier series of the function obtained by passing the voltage \(v(t)=V_{0} \cos 100 \pi t\) through a half-wave rectifier.