# PROBLEM SETS

## Problem Set 1 ( due Sep 12)

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

## Problem Set 2 ( due Sep 19)

- Consider the differential equation y” − x
^{2}y = 0.- Determine the dimension of each of the terms in this equation. (See the last paragraph on p.203 for definition.)
- Find the general solution of this equation in a Maclaurin series. Why are the coefficients of this series expressible in closed forms?

- Let y satisfy the equation

y” + xy = 0

and the initial conditions y(x_{0}) = 1, y'(x_{0}) = 0 where x_{0}> 0.- Find the WKB approximation of y(x) for x > x
_{0}. For what values of x do you expect it be a good approximation? Use the computer to obtain the numerical values of y for x

_{0}= 1, 5, 10. Compute also the numerical values of its WKB solution. Plot these two solutions together. How well do they agree? - Find the WKB approximation of y(x) for x > x
- Problem 4 in Chapter 7.

## Problem Set 3 ( due Sep 26)

- Consider the solution of the differential equation

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

for $\lambda=1,3,5,10$.

- Chapter 7, Prob 10.

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

- 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 $\lambda$.
- Chapter 8, Prob 5.
- Chapter 8, Prob 7 ( skip 3c)

## Problem Set 5 (due Oct 12)

- Prob 8 in Chapter 8.

## Problem Set 6 (due Oct 24)

- 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.
- Find two independent solutions of this differential equation in the forms of Laplace integral representation.
- Find the asymptotic forms of these two solutions as $x$ becomes very large, both positively and negatively.

- Evaluate the integral $\int_{-\infty}^{\infty}e^{ix^{4}}dx.$
- Let $I(\lambda)=\int_{0}^{\infty}e^{i\lambda x-1/x}dx,$ $\ \ \ \lambda>>1.$
- Calculate the leading term of this integral with the method of steepest descent.
- Justify your result by deforming the contour of integration into one on which the integrand is the largest at the saddle point.

## Problem Set 7 (due Oct 31)

- 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. - 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. - 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 $\dfrac{\lambda}{1+t}\overset{\cdot}{x}$.)

## Problem Set 8 (due Nov 7)

- 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 $\dfrac{\lambda}{1+t}\overset{\cdot}{x}$.) - 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. - Chapter 9, Prob 4.

## Problem Set 9 (due Nov 21)

- 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

$\Gamma(2z)=\dfrac{2^{2z-1}\Gamma(z)\Gamma(z+1/2)}{\sqrt{\pi}}.$ - Chapter 9, Prob 4.
- Chapter 9, Prob 5.

## Problem Set 10 (due Dec 5)

- 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 $\overset{\cdot}{x}(0)=A,$ $x(0)=0,$ $X(0)=B,$ $\overset{\cdot}{X}(0)=0.$- 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.
- Find $x_{0}(t)$ and $X_{0}(t).$
- Derive the differential equations satisfied by $x_{1}(t)$ and $X_{1}(t)$. Identify the secular terms in these equations.
- Find the renormalized angular frequency for these oscillators.