Differential Equations - Packet 11

Problems due Wednesday 3 Apr 2024 by 11:59pm

Easier Problems

Problem 11-01

Laplace transforms

Find the Laplace transforms of the following functions:

• (a) $\sin (at)$ (do the derivation, don’t just cite the chart)
• (b) $4t\sin (t)\cos (t)$ (hint: trig identities)

Problem 11-02

Inverse Laplace transforms

Find the inverse Laplace transforms of the following functions:

• (a) $\frac{p}{4p^2+1}$
• (b) $\frac{2p-3}{p^2-4}$
• (c) $\frac{p+3}{p^2+2p+5}$

Harder Problems

Problem 11-03

Periodic functions

Suppose that $f(t)$ is periodic with period $T$. This means that $f$ satisfies the equation $f(t+T)=f(t)$ for any $t$.

Notice that this equation implies:

$$\begin{array}{cc}f(t+T)& =f(t)\\ f(t+2T)& =f((t+T)+T)=f(t+T)=f(t)\\ f(t+3T)& =f(t)\\ ⋮& \\ f(t+nT)& =f(t),\text{any }n.\end{array}$$

So the value of $f$ at any point can be traced back to the values in the range $t\in [0,T)$.

Show that for such $f$, its Laplace transform is given by:

$$ℒ\{f\}=\frac{{\displaystyle {\int }_0^T{e}^{-pt}f(t)dt}}{1-e^{-pT}}.$$

Problem 11-04

Ramp drive

A ramp function is given by the piecewise function $g(t)$ defined for some $t_0>0$:

$$g(t)=\{\begin{array}{cc}mt& 0\le t\le t_0\\ 0& t>t_0.\end{array}$$

This function may be written using the Heaviside step function as follows: $g(t)=mt(u_0(t)-u_{t_0}(t))$.

• (a) Find the Laplace transform $ℒ\{u_c(t)\}$ of the step function.
• (b) Relate the result of (a) to the shift operation from lecture: $ℒ\{e^{at}f(t)\}=F(p-a)$.
• (c) Find the Laplace transform $ℒ\{g(t)\}$ of the ramp function using the result of (a).
• (d) Now solve the differential equation: $y^{\prime \prime }+4y=g(t)$ with $y(0)=y^{\prime }(0)=0$.
• (e) [extra credit] Find the Laplace transform of the sawtooth function which is a sequence of consecutive ramps: $g(t)=2t$ for $0\le t<\pi$ and $g(t+\pi )=g(t)$. (Hint: Use Problem 11-03.)