91Ó°ÊÓ

Let \(f: \mathbf{R} \rightarrow \mathbf{R}\) be given by \(f(x)=1-x^{2}\) if \(-1

Find the derivatives \(f^{\prime}\) and \(f^{\prime \prime}\), if \(f(t)=\left|t^{3}-t\right|\). Sketch the graphs of \(f\), \(f^{\prime}\) of \(f^{\prime \prime}\) in separate pictures.

Find the general solution of the differential equation \(\frac{d y}{d t}+2 t y=\delta(t-a)\).

Find \(y=y(x)\) that satisfies \(\left(1+x^{2}\right) y^{\prime}-2 x y=\delta(x-1)\) and \(y(0)=1\)

Show that the function \(f(z)=e^{4 z}\) has period \(\pi i / 2\).

Let \(f\) be a continuous function on \(\mathbf{R}\). Assume that we know that it has period \(2 \pi\) and that it satisfies the equation $$ f(t)=\frac{1}{2}\left(f\left(t-\frac{1}{2} \pi\right)+f\left(t+\frac{1}{2} \pi\right)\right) \quad \text { for all } t \in \mathbf{R} \text {. } $$ Show that \(f\) in fact has a period shorter than \(2 \pi\), and determine this period.

Let \(\varphi\) be a \(C^{1}\) function such that \(\varphi\) and \(\varphi^{\prime}\) are bounded on the real axis. Compute the limit $$ \lim _{n \rightarrow \infty} \frac{2 n^{3}}{\pi} \int_{-\infty}^{\infty} \frac{x}{\left(1+n^{2} x^{2}\right)^{2}} \varphi(x) d x $$

Let \(F(x)=\left(1-x^{2}\right)(H(x+1)-H(x-1))\). Let \(g\) be continuous on the interval \([-1,1]\). Find the limit $$ \lim _{n \rightarrow \infty} \frac{3}{4} \int_{-1}^{1} n F(n x) g(x) d x . $$

Find \(y(x)\) that solves the differential equation \(y^{\prime}+\frac{x^{2}+1}{x} y=\delta(x-2)\) and satisfies \(y(1)=1\).

Let \(f: \mathbf{R} \rightarrow \mathbf{R}\) be described by \(f(x)=\left(x^{2}-1\right)^{2}(H(x+1)-H(x-1))\). Show that \(f\) belongs to the class \(C^{1}\) but not to \(C^{2}\). Also compute its third derivative.