91Ó°ÊÓ

Solve $$ \begin{array}{l} y_{t}+2 y_{x}=x+t, \quad 00, \\ y(0, t)=0, \quad y(x, 0)=0. \end{array} $$

Derive the first shifting property from the definition of the Laplace transform.

Suppose that \(f(t)\) and \(g(t)\) are differentiable functions and suppose that \(f(t)=$$g(t)=0\) for all \(t \leq 0\). Show that $$ (f * g)^{\prime}(t)=\left(f^{\prime} * g\right)(t)=\left(f * g^{\prime}\right)(t) $$.

Find the solution to $$ m x^{\prime \prime}+c x^{\prime}+k x=f(t), \quad x(0)=0, \quad x^{\prime}(0)=0 $$ for an arbitrary function \(f(t),\) where \(m>0, c>0, k>0,\) and \(c^{2}-4 k m<0\) (system is underdamped). Write the solution as a definite integral.

For an \(\alpha>0,\) solve $$ \begin{array}{l} y_{t}+\alpha y_{x}+y=0, \quad 00, \\ y(0, t)=\sin (t), \quad y(x, 0)=0. \end{array} $$

Using the Laplace transform solve $$ m x^{\prime \prime}+c x^{\prime}+k x=0, \quad x(0)=a, \quad x^{\prime}(0)=b, $$ where \(m>0, c>0, k>0,\) and \(c^{2}-4 k m<0\) (system is underdamped).

Find the Laplace transform of \(3+t^{5}+\sin (\pi t)\).

Find the solution to $$ m x^{\prime \prime}+c x^{\prime}+k x=f(t), \quad x(0)=0, \quad x^{\prime}(0)=0 $$ for an arbitrary function \(f(t),\) where \(m>0, c>0, k>0,\) and \(c^{2}=4 k m\) (system is critically damped). Write the solution as a definite integral.

Using the Laplace transform solve $$ m x^{\prime \prime}+c x^{\prime}+k x=0, \quad x(0)=a, \quad x^{\prime}(0)=b, $$ where \(m>0, c>0, k>0,\) and \(c^{2}=4 \mathrm{~km}\) (system is critically damped).

Suppose that \(L x=\delta(t), x(0)=0, x^{\prime}(0)=0,\) has the solution \(x=e^{-t}\) for \(t>0\). Find the solution to \(L x=t^{2}, x(0)=0, x^{\prime}(0)=0\) for \(t>0\).