91Ó°ÊÓ

Consider the initial value problem (involving Bessel's equation of order 0) \(t y^{\prime \prime}+\) \(y^{\prime}+t y=0, y(0)=1, y^{\prime}(0)=0\), with solution \(J_{0}(t)\), the Bessel function of order 0 . Use Laplace transforms to show that \(\mathcal{L}\\{y(t)\\}=\mathcal{L}\left\\{J_{0}(t)\right\\}=k / \sqrt{s^{2}+1}\), where \(k\) is a constant. (Hint: \(\mathcal{L}\left\\{t y^{\prime \prime}\right\\}=-d / d s\left(s^{2} Y(s)-s\right)\) and \(\mathcal{L}\\{t y\\}=-d / d s Y(s)\).)

(a) Use the linearity of the Laplace transform to compute \(\mathcal{L}\\{a \sin b t-b \sin a t\\}\) and \(\mathcal{L}\\{\cos b t-\cos a t\\}\). (b) Use these results to find \(\mathcal{L}^{-1}\left\\{1 /\left[\left(s^{2}+a^{2}\right)\left(s^{2}+b^{2}\right)\right]\right\\}\) and \(\mathcal{L}^{-1}\\{s /\) \(\left.\left[\left(s^{2}+a^{2}\right)\left(s^{2}+b^{2}\right)\right]\right\\}\).

\(y^{\prime \prime}+4 y^{\prime}+13 y=e^{-2 t} \cos 3 t+1, y(0)=y^{\prime}(0)\) \(=1\)

Show that the convolution integral is associative by proving that \((f *(g * h))(t)=\) \(((f * g) * h)(t)\).

Suppose that an object with mass \(m=1\) is attached to the end of a spring with spring constant 16. If there is no damping and the spring is subjected to the forcing function \(f(t)=\sin t\), determine the motion of the spring if at \(t=1\), the spring is supplied with an upward shock of 4 units.

(a) Use the definition of exponential order to show that if \(\lim _{t \rightarrow \infty} f(t) e^{-b t}\) exists and is finite, \(f(t)\) is of exponential order and if \(\lim _{t \rightarrow \infty} f(t) e^{-b t}=\infty\) for every value of \(b\), \(f(t)\) is not of exponential order. (b) Give an example of a function \(f(t)\) of exponential order \(b\) for which \(\lim _{t \rightarrow \infty} f(t) e^{-b t}\) does not exist.

\(x^{\prime \prime}+4 x=f(t), f(t)=\left\\{\begin{array}{l}t-1,0 \leq t<1 \\\ 0,1 \leq t<2\end{array}\right.\) and \(f(t)=f(t-2)\) if \(t \geq 2, x(0)=x^{\prime}(0)=0\)