12-267/Homework Assignment 2

DBN: Classes: 2012-13: 12-267 / Navigation Additions to this web site no longer count towards good deed points. # Week of... Notes and Links 1 Sep 10 About This Class.  Monday: Introduction and the Brachistochrone.  Tuesday: More on the Brachistochrone, administrative issues. Tuesday Notes.  Friday: Some basic techniques: first order linear equations. 2 Sep 17  Monday: Separated equations, escape velocities. HW1.  Tuesday: Escape velocities, changing source and target coordinates, homogeneous equations.  Friday: Reverse engineering separated and exact equations. 3 Sep 24  Monday: Solving exact equations, integration factors. HW2.  Tuesday: Statement of the Fundamental Theorem. Class Photo.  Friday: Proof of the Fundamental Theorem. 4 Oct 1  Monday: Last notes on the fundamental theorem. HW3.  Tuesday Hour 1: The chain law, examples of variational problems.  Tuesday Hour 2: Deriving Euler-Lagrange.  Friday: Reductions of Euler-Lagrange. 5 Oct 8 Monday is thanksgiving.  Tuesday: Lagrange multiplyers and the isoperimetric inequality. HW4.  Friday: More Lagrange multipliers, numerical methods. 6 Oct 15  Monday: Euler and improved Euler.  Tuesday: Evaluating the local error, Runge-Kutta, and a comparison of methods.  Friday: Numerical integration, high order constant coefficient homogeneous linear ODEs. 7 Oct 22  Monday: Multiple roots, reduction of order, undetermined coefficients.  Tuesday: From systems to matrix exponentiation. HW5. Term Test on Friday. 8 Oct 29  Monday: The basic properties of matrix exponentiation.  Tuesday: Matrix exponentiation: examples.  Friday: Phase Portraits. HW6. Nov 4 was the last day to drop this class 9 Nov 5  Monday: Non-homogeneous systems.  Tuesday: The Catalan numbers, power series, and ODEs.  Friday: Global existence for linear ODEs, the Wronskian. 10 Nov 12 Monday-Tuesday is UofT November break. HW7.  Friday: Series solutions for $y'=f(x,y)$. 11 Nov 19  Monday: $\pi$ is irrational, more on the radius of convergence.  Tuesday (class): Airy's equation, Fuchs' theorem.  Tuesday (tutorial): Regular singular points. HW8.  Friday: Discussion of regular singular points.. 12 Nov 26  Monday: Frobenius series by computer. Qualitative Analysis Handout (PDF).  Tuesday: The basic oscillation theorem. Handout on the Frobenius Method. HW9.  Friday: Non-oscillation, Sturm comparison. 13 Dec 3  Monday: More Sturm comparisons, changing the independent variable.  Tuesday: Amplitudes of oscillations. Last class was on Tuesday! F1 Dec 10 F2 Dec 17 The Final Exam (time, place, style, office hours times) Register of Good Deeds Add your name / see who's in!

This assignment is due at the tutorial on Tuesday October 2. Here and everywhere, neatness counts!! You may be brilliant and you may mean just the right things, but if the your readers will be having hard time deciphering your work they will give up and assume it is wrong.

Task 0. Identify yourself in the Class Photo!

Task 1. Solve the following differential equations:

1. $x^{2}y^{3}+x(1+y^{2})y'=0$ (hint: try $\mu =x^{\alpha }y^{\beta }$).
2. $dx+({\frac {x}{y}}-\sin y)dy=0$.
3. $(x^{2}+3xy+y^{2})dx-x^{2}dy=0$.
4. ${\frac {dy}{dx}}={\frac {2y-x+5}{2x-y-4}}$ (hint: consider trying $x_{1}=x+\alpha$ and $y_{1}=y+\beta$ for good $\alpha$, $\beta$).
5. $y'={\frac {y^{3}}{1-2xy^{2}}}$ with $y(0)=1$.
6. ${\frac {dy}{dx}}={\frac {2y+{\sqrt {x^{2}-y^{2}}}}{2x}}$.

Task 2. Let $M$ and $N$ be differentiable functions of $x$ and $y$.

1. Show that if ${\frac {N_{x}-M_{y}}{xM-yN}}$ depends only on $xy$, then the differential equation $Mdx+Ndy=0$ has an integrating factor of the form $\mu (xy)$, where $\mu$ is a function of a single variable.
2. Find a condition on $M$ and $N$ that would imply that the differential equation $Mdx+Ndy=0$ would have an integrating factor of the form $\mu (x+y)$, where $\mu$ is a function of a single variable.

Task 3. The equation $y'+p(x)y=q(x)y^{n}$ is called a "Bernoulli Equation".

1. Explain why you already know how to solve the Bernoulli equation when $n=0$ and when $n=1$.
2. Show that if $n\neq 0,1$, then the substitution $v=y^{1-n}$ reduces the Bernoulli equation to an equation you already know how to solve.
3. Solve the equation $x^{2}y'+2xy-y^{3}=0$ (for $x>0$).

Task 4. Find an example of a non-differentiable function which is nevertheless Lipschitz.

Dror's notes above / Student's notes below

Solution to Task 2 --Twine 18:44, 24 October 2012 (EDT)

Part 1

What we want is an integration factor $\mu$ such that $(\mu M)_{y}=(\mu N)_{x}$. Let $xy=z$

When $\mu$ is a function of z, ${\frac {dv}{dx}}={\frac {dv}{dz}}{\frac {dz}{dx}}={\frac {dv}{dz}}y$.

Similarly ${\frac {dv}{dy}}={\frac {dv}{dz}}x$.

Then we have

$(\mu M)_{y}=(\mu N)_{x}$

$\iff \mu _{z}xM-\mu _{xy}yN=\mu (N_{x}-M_{y})$

$\iff {\frac {\mu _{z}}{mu}}={\frac {N_{x}-M_{y}}{xM-yN}}$

If the right hand side depends only on xy, we can get

$\mu (xy)=e^{\int {\frac {N_{x}-M_{y}}{xM-yN}}d(xy)}$

which satisfies the requirements of an integrating factor.

Part 2

As in Part 1, we need $(\mu M)_{y}=(\mu N)_{x}$. Let $z=x+y$

When $\mu$ is a function of $x+y$, ${\frac {d\mu }{dx}}={\frac {d\mu }{dz}}{\frac {dz}{dx}}={\frac {d\mu }{dz}}$.

Similarly ${\frac {d\mu }{dy}}={\frac {d\mu }{dz}}$

Then we have

$(\mu M)_{y}=(\mu N)_{x}$

$\iff {\frac {\mu _{z}}{\mu }}={\frac {N_{x}-M_{y}}{M-N}}$

If the right hand side of this equation depends only on z (that is, only on (x+y)), then we have

$\mu (x+y)=e^{\int {\frac {N_{x}-M_{y}}{M-N}}d(x+y)}$

which satisfies the requirements of an integrating factor.

Solution to HW2: Mathstudent

• 

  page 1

• 

  page 2

• 

  page 3

• 

  page 4

• 

  page 5

• 

  page 6

• 

  page 7