MATH528

MATH528Due date: 09/20/18

Homework 03

Note: This will be the last homework in which model solutions are presented before due date. At this point, enough familiarity with TeXmacs and Mathematica has been obtained for students to complete subsequent homeworks.

1Exercises

Exercise. RQ.1.22

Solution. $y^{'} + 4 x y = e^{- 2 x^{2}}$ is a linear ODE of form $y^{'} + p (x) y = r (x)$ , with $p (x) = 4 x$ , $r (x) = e^{- 2 x^{2}}$ , with solution

y (x) = e^{- h (x)} (\int e^{h (x)} r (x) d x + c), h (x) = \int p (x) d x .

Compute:

h (x) = 2 x^{2}

\int e^{h (x)} r (x) d x = \int d x = x,

hence

y (x) = e^{- 2 x^{2}} (x + c),

and $y (0) = - 4.3$ gives $c = - 4.3$ . Verify:

In[1]:=

ODE = y'[x]+4 x y[x] == Exp[-2x^2]

$y^{'} (x) + 4 x y (x) = e^{- 2 x^{2}}$

In[2]:=

iCond = y[0]==-4.3

$y (0) = - 4.3$

In[3]:=

DSolve[{ODE,iCond},y[x],x]

${{y (x) \to e^{- 2 x^{2}} (x - 4.3)}}$

In[4]:=

Exercise. PS1.1.6

Solution.

In[1]:=

Exercise. PS1.1.7

Solution.

In[1]:=

Exercise. PS1.1.8

Solution.

In[1]:=

Exercise. PS1.2.2

Solution.

In[1]:=

Exercise. PS1.2.3

Solution.

In[1]:=

Exercise. PS1.2.4

Solution.

In[1]:=

Exercise. PS1.3.5

Solution.

In[1]:=

2Problems

Problem. RQ.1.27

Solution. The bacteria population $y (t)$ is given by $y^{'} = r y$ , $y (0) = y_{0}$ , with solution $y (t) = e^{r t} y_{0}$ . After 1 day, $y (1) = e^{r} y_{0} = 1.25 y_{0} \Rightarrow r = \ln 1.25$ . Population double condition is $e^{r t} = 2 \Rightarrow r t = \ln 2 \Rightarrow t = \ln 2 / \ln 1.25$ . Population triple condition is $e^{r t} = 3 \Rightarrow r t = \ln 3 \Rightarrow t = \ln 3 / \ln 1.25$ . Verify:

In[6]:=

ODE = y'[t] == r y[t]

$y^{'} (t) = r y (t)$

In[5]:=

iCond = y[0] == y0

$y (0) = y0$

In[9]:=

sol[t_,r_] = y[t] /. DSolve[{ODE,iCond},y[t],t][[1,1]]

$y0 e^{r t}$

In[11]:=

{sol[1,Log[1.25]]/y0,

sol[Log[2]/Log[1.25],Log[1.25]]/y0,

sol[Log[3]/Log[1.25],Log[1.25]]/y0}

${1.25, 2 ., 3 .}$

In[12]:=

Problem. PS1.1.17

Solution.

In[1]:=

Problem. PS1.1.18

Solution.

In[1]:=

Problem. PS1.3.22

Solution.

In[1]:=

3Projects

3.1PS1.7.6

See Lesson 4, slide 3 for an example of Picard's iteration.