Problem 15
Use the Laplace transform to solve the given initial value problem. $$ y^{\prime \prime}-2 y^{\prime}-2 y=0 ; \quad y(0)=2, \quad y^{\prime}(0)=0 $$
Problem 15
Find an expression involving \(u_{c}(t)\) for a function \(g\) that ramps up from zero at \(t=t_{0}\) to the value \(h\) at \(t=t_{0}+k\) and then ramps back down to zero at \(t=t_{0}+2 k\)
Problem 15
Find the inverse Laplace transform of the given function. $$ F(s)=\frac{2(s-1) e^{-2 s}}{s^{2}-2 s+2} $$
Problem 15
Express the solution of the given initial value problem in terms of a convolution integral. \(y^{\prime \prime}+y^{\prime}+\frac{5}{4} y=1-u_{\pi}(t) ; \quad y(0)=1, \quad y^{\prime}(0)=-1\)
Problem 15
Using integration by parts, find the Laplace transform of the given function; \(n\) is a positive integer and \(a\) is a real constant. $$ t e^{a t} $$
Problem 16
Consider the initial value problem $$ y^{\prime \prime}+y=f_{k}(t), \quad y(0)=0, \quad y^{\prime}(0)=0 $$ where \(f_{k}(t)=\left[u_{4-k}(t)-u_{4+k}(t)\right] / 2 k\) with \(0< k \leq 1\) (a) Find the solution \(y=\phi(t, k)\) of the initial value problem. (b) Calculate lim \(\phi(t, k)\) from the solution found in part (a). (c) Observe that \(\lim _{k \rightarrow 0} f_{k}(t)=\delta(t-4)\). Find the solution \(\phi_{0}(t)\) of the given initial value problem with \(f_{k}(t)\) replaced by \(\delta(t-4) .\) Is it true that \(\phi_{0}(t)=\lim _{k \rightarrow 0} \phi(t, k) ?\) (d) Plot \(\phi(t, 1 / 2), \phi(t, 1 / 4),\) and \(\phi_{0}(t)\) on the same axes. Describe the relation between \(\phi(t, k)\) and \(\phi_{0}(t) .\)
Problem 16
Use the Laplace transform to solve the given initial value problem. $$ y^{\prime \prime}+2 y^{\prime}+5 y=0 ; \quad y(0)=2, \quad y^{\prime}(0)=-1 $$
Problem 16
Using integration by parts, find the Laplace transform of the given function; \(n\) is a positive integer and \(a\) is a real constant. $$ t \sin a t $$
Problem 16
A certain spring-mass system satisfies the initial value problem $$ u^{\prime \prime}+\frac{1}{4} u^{\prime}+u=k g(t), \quad u(0)=0, \quad u^{\prime}(0)=0 $$ where \(g(t)=u_{3 / 2}(t)-u_{5 / 2}(t)\) and \(k>0\) is a parameter. (a) Sketch the graph of \(g(t)\). Observe that it is a pulse of unit magnitude extending over one time unit. (b) Solve the initial value problem. (c) Plut the solutition fro \(k=1 / 2, k=1,\) and \(k=2 .\) Describe the principal features of the solution and how they depend on \(k .\) (d) Find, to two decimal places, the smallest value of \(k\) for which the solution \(u(t)\) reaches the vilue? (e) Suppose \(k=2 .\) Find the time \(\tau\) after which \(|u(t)|<0.1\) for all \(t>\tau\)
Problem 16
Find the inverse Laplace transform of the given function. $$ F(s)=\frac{2 e^{-2 s}}{s^{2}-4} $$
