EMA 548 HW1 Spring 2014, University of Wisconsin, Madison

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EMA 548 HW1 Spring 2014, University of Wisconsin, Madison

Nasser M. Abbasi

November 28, 2019

1 Problem 1 (BO section 3.1, 3.3,3.4)
1.1 problem 3.3 (c)
1.2 problem 3.3 (h)
1.3 problem 3.4 (d)
1.4 problem 3.4 (f)
2 Problem 2 (BO section 3.2, 3.7,3.8)
2.1 Problem 3.7
2.2 Problem 3.8
3 Problem 3 (BO section 3.3, 3.24f)
4 Problem 4 (BO section 3.4, 3.27,3.33(b,f)
4.1 Problem 3.27
4.2 Problem 3.33b
4.3 Problem 3.33f
5 Problem 5 (BO section 3.5, 3.39(b),3.46(a)
5.1 Problem 3.39(b)
5.2 Problem 3.46(a)
6 Problem 6 (BO section 3.6, 3.49(b))

1 Problem 1 (BO section 3.1, 3.3,3.4)

1.1 problem 3.3 (c)

problem: Classify all the singular points (finite and infinite) of the following

x (1 − x )y′′ + (c − (a + b + 1)x )y′ − aby = 0

Answer:

Writing the DE in standard form

$x = 0,1$ are singular points. To classify what type of singularity, looking at $x = 0$ then

Hence, $limx →0 xp (x ) → c$ and $limx →0 x2q (x) → 0$ , therefore the singularity at $x = 0$ is removable, hence $x = 0$ is a reqular singular point.

Now, looking at $x = 1$ .

$limx →1 (x − 1 )p(x ) → (− c + (a + b + 1))$ and $2 limx →1 (x − 1) q(x) → 0$ , therefore the singularity at $x = 1$ is also removable, hence $x = 1$ is a reqular singular point.

To check the type of singularity, if any, at $x = ∞$ , the DE is first transformed using

1- x = t

(1)

This uses

-d- d--dt 2 d dx = dtdx = − t dt

(2)

and

Sustituting eqs (1,2,3) into the original DE gives

Writing the above in standard form

d2y (2t(t − 1) − t2c + (a + b + 1) t)dy ab dt2-+ -----------t2(t −-1-)----------dt-− t2-(t-−-1)y = 0

Expanding

◜------------p◞(t◟)-----------◝ ◜--q◞(◟t)--◝ d2y ( 2 c (a + b + 1 )) dy ab ----+ --− -------+ ----------- ---− ---------y = 0 dt2 t (t − 1) t (t − 1) dt t2(t − 1)

Hence at $t = 0$ there is a singularity (this means $x = ∞$ ). To find what type

And

( ) ( ) lim t2q (t) = lim t2 − ----ab--- = lim --− ab- = ab t→0 t→0 t2 (t − 1) t→0 (t − 1)

Hence the singularity is removable. Therefore $x → ∞$ is a regular singular point.

1.2 problem 3.3 (h)

problem: Classify all the singular points (finite and infinite) of the following

( ) ( ( ) μ2 ) 1 − x2 y′′ − 2xy′ + λ + 40 1 − x2 − ------2- y = 0 (1 − x )

Answer:

Writing the DE in standard form

( ) ( ( ) μ2 ) 1 − x2 y′′ − 2xy′ + λ + 40 1 − x2 − ------2- y = 0 (1 − x )

1.3 problem 3.4 (d)

problem: Classify $x = 0$ and $x = ∞$ of the following

2 ′′ 1x x y = ye

Answer:

1.4 problem 3.4 (f)

problem: Classify $x = 0$ and $x = ∞$ of the following

′′ y = y ln x

Answer:

2 Problem 2 (BO section 3.2, 3.7,3.8)

2.1 Problem 3.7

Problem: Estimate the number of terms in the Taylor series (3.2.1) and (3.2.2) that are necessary to compute

$Ai (x)$ and $Bi (x)$ correct to three decimal places at $x = ±1, ±100, ±10000$

Answer:

2.2 Problem 3.8

Problem: How many terms in the Taylor series solution to $y′′′ = x3y$ with $y (0) = 1,y′(0) = y′′(0) = 0$ are needed to evaluate $∫ 1y (x)dx 0$ correct to three decimal places?

Answer:

3 Problem 3 (BO section 3.3, 3.24f)

Problem:

Find series expansions of all the solutions to the following differential equations about $x = 0$ . Try

to sum in closed form any infinite series that appear.

(sin x)y ′′ − 2 (cosx) y′ − (sinx) y = 0

Answer:

4 Problem 4 (BO section 3.4, 3.27,3.33(b,f)

4.1 Problem 3.27

Derive (3.4.28). Where 2.4.28 is the solution of example 5 which is stated here:

Local behavior of solutions near an irregular singular point of a general nth-order Schrodinger equation. In this example we derive an extremely simple and important formula for the leading behavior of solutions to the nth-order Schrodinger equation

dny- dxn = Q (x) y

near an irregular singular point at $x0.$

The exponential substitution $y = eS (x)$ and the asymptotic approximations $dkS-≪ (S ′)k dxk$ as $x → x0$ for $k = 2,3,⋅⋅⋅,n$ give the asymptotic differential equation $′n (S ) ∼ Q (x )$ $(x → x0)$ . Thus, $∫ 1 S (x ) ∼ ω xQ (t)n dt$ $(x → x0)$ $(x → x0 )$ , where $ω$ is an nth root of unity. This result determines the $n$ possible controlling factors of $y (x)$ . The leading behavior of $y(x)$ is found in the usual way (see Prob. 3.27) to be

( ∫ x ) y(x) ∼ cQ (x) 1−2nn-exp ω Q (t)1n dt ,(x → x ) 0

Answer:

4.2 Problem 3.33b

Find the leading behaviors as $+ x → 0$ of the following equations

x4y′′′ − 3x2y′ + 2y = 0

Answer:

4.3 Problem 3.33f

Find the leading behaviors as $x → 0+$ of the following equations

x4y′′ − x2y′ + 1y = 0 4

Answer:

5 Problem 5 (BO section 3.5, 3.39(b),3.46(a)

5.1 Problem 3.39(b)

Find the leading asymptotic behaviors as $x → ∞$ of the following equations

′′′ ′ xy = y

Answer:

5.2 Problem 3.46(a)

What is the leading behavior of solutions to $′′ -y′- y- y + x32 − x2 = 0$ as $x → ∞$ ? Show that it is inconsistent to assume that $S′′ ≪ S′2(x → ∞ ).$ However, show that the approximate equation $S ′′ + (S ′)2 ∼-1 (x → ∞ ) x2$ can be solved exactly by assuming a solution of the form $′ c S = x$