18.305 - Advanced Analytic Methods (Fall 2011)

# PROBLEM SETS

## Problem Set 1 ( due Sep 12)

1. Chapter 1, Prob 1a,1c;
2. Chapter 1, Prob 2b, 2d, 2f.

>> Solutions for PS1

## Problem Set 2 ( due Sep 19)

1. Consider the differential equation y” − x2y = 0.
1. Determine the dimension of each of the terms in this equation. (See the last paragraph on p.203 for definition.)
2. Find the general solution of this equation in a Maclaurin series. Why are the coefficients of this series expressible in closed forms?
2. Let y satisfy the equation
y” + xy = 0
and the initial conditions y(x0) = 1, y'(x0) = 0 where x0 > 0.
1. Find the WKB approximation of y(x) for x > x0. For what values of x do you expect it be a good approximation?
2. Use the computer to obtain the numerical values of y for x0 = 1, 5, 10. Compute also the numerical values of its WKB solution. Plot these two solutions together. How well do they agree?
3. Problem 4 in Chapter 7.

>> Solutions for PS2

## Problem Set 3 ( due Sep 26)

1. Consider the solution of the differential equation
$\dfrac{d^{2}y}{dx^{2}}+\lambda^{2}x^{4}y=0,$ satisfying $y(1)=0,$ $y^{\prime}(1)=1.$
1. Find its WKB approximation (7.8) and compare it numerically with its exact solution for $\lambda=1,3,5,10$.
2. Find its WKB approximation with its second order term included and compare it numerically with it exact solution
for $\lambda=1,3,5,10$.
2. Chapter 7, Prob 10.

>> Solutions for PS3

## Problem Set 4 (due Oct 3, Monday)

1. Chapter 8, Problem 3. Compute also the exact values of these integrals and compare them with the numerical values of their leading terms as a function of $\lambda$.
2. Chapter 8, Prob 5.
3. Chapter 8, Prob 7 ( skip 3c)

>> Solutions for PS4

## Problem Set 5 (due Oct 12)

1. Prob 8 in Chapter 8.

>> Solutions for PS5

## Problem Set 6 (due Oct 24)

1. Consider the differential equation $\dfrac{d^{2}y}{dx^{2}}+x\dfrac{dy}{dx}+(n+1)y=0,$ where $n$ is a non-negative integer.
1. Find two independent solutions of this differential equation in the forms of Laplace integral representation.
2. Find the asymptotic forms of these two solutions as $x$ becomes very large, both positively and negatively.
2. Evaluate the integral $\int_{-\infty}^{\infty}e^{ix^{4}}dx.$
3. Let $I(\lambda)=\int_{0}^{\infty}e^{i\lambda x-1/x}dx,$ $\ \ \ \lambda>>1.$
1. Calculate the leading term of this integral with the method of steepest descent.
2. Justify your result by deforming the contour of integration into one on which the integrand is the largest at the saddle point.

>> Solutions for PS6

## Problem Set 7 (due Oct 31)

1. Find the leading asymptotic form of the integral
$I(\lambda)=\int_{-\infty}^{\infty}e^{i\lambda x}e^{ix^{4}}dx,$ $\ \lambda >>1.$
Compare this asymptotic form with the numerical value of the integral.

2. Find the leading asymptotic form of the integral
$I(\lambda)=\int_{-1}^{1}e^{i\lambda x}e^{-1/(1-x^{2})}dx,$ $\ \lambda>>1.$
Compare this asymptotic form with the numerical value of the integral.

3. Find the approximate solution of
$\overset{\cdot\cdot}{x}+\dfrac{\lambda}{1+t}\overset{\cdot}{x}+x=0,$ $\lambda>>1,$
satisfying
$x(0)=1,$ $\overset{\cdot}{x}(0)=0.$ In the above, $\overset{\cdot }{x}\equiv\dfrac{dx}{dt}.$
(The equation above holds for a harmonic oscillator under a large and time-dependent frictional force $\dfrac{\lambda}{1+t}\overset{\cdot}{x}$.)

>>Solutions for PS7

## Problem Set 8 (due Nov 7)

1. Find the approximate solution of $\overset{\cdot\cdot}{x}+\dfrac{\lambda}{1+t}\overset{\cdot}{x}+x=0,$ $\lambda>>1,$ satisfying $x(0)=1,$ $\overset{\cdot}{x}(0)=0.$ In the above, $\overset{\cdot }{x}\equiv\dfrac{dx}{dt}.$
(The equation above holds for a harmonic oscillator under a large and time-dependent frictional force $\dfrac{\lambda}{1+t}\overset{\cdot}{x}$.)
2. Find the solution of
$\epsilon y"-(1+x)y^{\prime}+y=0,$   $0 < x < 1,$   $y(0)=a,$   $y(1)=b,$
accurate up to ( and including) the order $\epsilon.$ Find the numerical solution for this problem with $a=b=1$ and compare graphically with the approximate solution you obtain.
3. Chapter 9, Prob 4.

>> Solutions for PS8

## Problem Set 9 (due Nov 21)

1. The Bessel function $J_{p}(x)$ is given by the integral representation
$J_{p}(x)=\dfrac{x^{p}}{2^{p}\sqrt{\pi}\Gamma(p+1/2)}\int_{-1}^{1}$ $e^{-ix\rho}(1-\rho^{2})^{-1/2+p}d\rho.$
Deduce from the above expression the Maclaurin series for $J_{p}(x).$
Note: $\int_{0}^{1}t^{p-1}(1-t)^{q-1}dt=\dfrac{\Gamma(p)\Gamma(q)}% {\Gamma(p+q)},$ which is known as the Beta function $B(p,q).$

We also have
$\Gamma(2z)=\dfrac{2^{2z-1}\Gamma(z)\Gamma(z+1/2)}{\sqrt{\pi}}.$

2. Chapter 9, Prob 4.
3. Chapter 9, Prob 5.

>>Solutions for PS9

## Problem Set 10 (due Dec 5)

1. The motion of two coupled harminic oscillators, of coordinates $x$ and $X$ respectively, obeys the Euler-Lagrange equations derived from the Lagrangian
$L=\dfrac{\overset{\cdot}{x}^{2}+\text{ }\overset{\cdot}{X}^{2}}{2}% -\dfrac{\omega_{0}^{2}x^{2}+W_{0}^{2}X^{2}+\dfrac{\epsilon}{2}(x^{2}% +X^{2})^{2}}{2},$   $\omega_{0}\neq W_{0}.$
The initial conditions are $\overset{\cdot}{x}(0)=A,$ $x(0)=0,$ $X(0)=B,$ $\overset{\cdot}{X}(0)=0.$
1. Derive from the Lagrangian above the equations of motions for this dynamical system. Express $\omega_{0}$, $W_{0}$, $x(t)$, and $X(t)$ in these equations by their corresponding perturbation series.
2. Find $x_{0}(t)$ and $X_{0}(t).$
3. Derive the differential equations satisfied by $x_{1}(t)$ and $X_{1}(t)$. Identify the secular terms in these equations.
4. Find the renormalized angular frequency for these oscillators.

>>Solutions for PS10