Math · 07
Attempt these before you open a book or a video. Write the argument in the work box and the conclusion in the answer box. Check counts the problem only when both pass, and it does not open the solution. The solution stays locked for twenty minutes after you start the timer.
Attack 1
Solve \(y'=y\) with \(y(0)=1\).
Timer not started. Twenty minutes of real work before a solution.
Separable: \(dy/y=dt\), so \(\ln|y|=t+C\), hence \(y=Ae^{t}\). The condition \(y(0)=1\) gives \(A=1\). Or recognize that the derivative equaling the function characterizes the exponential.
Attack 2
Solve \(y'+y=0\).
Timer not started. Twenty minutes of real work before a solution.
\(y=Ce^{-t}\). Integrating factor \(e^{t}\) turns the left side into \((ye^{t})'\).
Attack 3
For \(y'=y(1-y)\), find the equilibria and say which are stable.
Timer not started. Twenty minutes of real work before a solution.
Equilibria at \(y=0\) and \(y=1\). For \(y\in(0,1)\) the right-hand side is positive, so arrows point up toward 1 and away from 0. For \(y>1\) it is negative, toward 1. For \(y<0\) it is negative, away from 0. So 1 is stable and 0 is unstable.
An ordinary differential equation asks for a function whose derivatives satisfy a relation. The first course has two layers. The computational layer is a list of solvable classes: separable equations, first-order linear equations via an integrating factor, second-order linear equations with constant coefficients, Laplace transforms, and power-series solutions at ordinary points. The structural layer is what you keep: existence and uniqueness, the phase line and the phase plane, linearization near equilibria, and the fact that an \(n\)th-order equation is a first-order system in a higher-dimensional space.
Arnold’s Ordinary Differential Equations is the ceiling. The MIT Press edition has ISBN 978-0-262-51018-9. It is paid, and it is the book whose problems are worth more than another technique chapter. Boyce and DiPrima is the standard paid course text; no edition was verified, so none is cited. Tenenbaum and Pollard, ISBN 978-0-486-64940-5, is the huge Dover problem collection. You will actually work MIT 18.03, which has the lectures, recitations, and exams.
A phase portrait you cannot sketch from the equation is a picture you copied. Sketch before you solve.
Statements only. Write the argument before you reveal it. A revealed proof you did not attempt is a paragraph you will not be able to rebuild at 2am.
First order. Separable: \(y'=f(x)g(y)\). Linear: \(y'+p(x)y=q(x)\) has integrating factor \(e^{\int p}\). Picard–Lindelöf: if \(f(t,y)\) is continuous and Lipschitz in \(y\) near \((t_0,y_0)\), the initial-value problem has a unique solution on some interval.
The integrating factor is chosen so the left side becomes a derivative. Picard iteration is the contraction mapping on a short-time space of continuous functions; you should be able to state the theorem now and prove the contraction estimate when you have analysis under your belt. Do not pretend the theorem is “just separation of variables.”
Second-order linear. For \(y''+ay'+by=0\), try \(e^{rt}\) and solve the characteristic polynomial. Repeated roots bring a factor of \(t\). A particular solution of the nonhomogeneous equation plus the homogeneous general solution is the general solution. Wronskian nonzero means the two homogeneous solutions form a basis.
The equation is linear, so the solution space of the homogeneous problem is a vector space; the existence theorem says the dimension equals the order. The characteristic polynomial is the computation. Variation of parameters produces a particular solution from the Wronskian.
Systems and phase planes. \(\mathbf x'=A\mathbf x\) is solved by the exponential \(e^{tA}\), computed via Jordan form or, when \(A\) is diagonalizable, via eigenvalues. Equilibria of \(\mathbf x'=\mathbf F(\mathbf x)\) are where \(\mathbf F=0\). The linearization \(D\mathbf F\) classifies a hyperbolic equilibrium. A sign chart or nullclines sketch the rest.
If \(Av=\lambda v\) then \(e^{\lambda t}v\) solves the system. The linearization theorem says a hyperbolic equilibrium looks like its linear part; the proof is Hartman–Grobman, which you may quote after you have checked the eigenvalues are off the imaginary axis.
Laplace transforms. \(\mathcal L\{y\}(s)=\int_0^\infty e^{-st}y(t)\,dt\). It turns \(y'\) into \(sY-y(0)\) and convolutions into products, under growth hypotheses that justify the integrals.
Integration by parts on the integral for \(y'\), with the boundary term at infinity killed by the exponential decay you assumed. Tables are not a proof. Derive the transforms of \(1\), \(e^{at}\), and \(\sin bt\) once.
Series and a first PDE. At an ordinary point, a linear ODE has a power-series solution whose radius is at least the distance to the nearest complex singularity of the coefficients. The heat equation \(u_t=u_{xx}\) and the wave equation \(u_{tt}=c^2 u_{xx}\), with boundary conditions, are solved by separating variables into an ODE eigenvalue problem.
The series method is plugging in \(\sum a_n x^n\) and matching coefficients; convergence needs a majorant or a citation of the existence theorem in the complex domain. Separation of variables is linear algebra: eigenfunctions of a symmetric differential operator form a basis you can expand the initial data in. You will prove the basis claim in a later course; here, do it for sine series on \([0,\pi]\) by quoting Fourier.
Four rungs, in order. Do not skip a rung because the next one looks more serious.
Watch or read once. Do not study from this.
Paul’s Online Math Notes, Differential Equations
Read once, do not study from this. It is a technique index. Knowing the names of the techniques is the preview. Computing them is the course.
The thing you work through.
MIT 18.03SC
Lectures, recitations, problem sets, exams. This is the main text. Boyce and DiPrima is the usual paid book; no edition was verified. Use 18.03 so the problem sets have published solutions.
The problem book.
Tenenbaum and Pollard, Ordinary Differential Equations, ISBN 978-0-486-64940-5 PAID
The Dover book is the problem mountain and it is paid; the ISBN is catalogued. The link is 18.03’s problem sets, which you can do tonight. Use Tenenbaum when you want more drills of a type 18.03 only assigned twice.
The hardest questions that exist at this level.
Arnold, Ordinary Differential Equations, ISBN 978-0-262-51018-9
Arnold is paid and is the ceiling: geometry of phase portraits, not another integrating factor. No free legal edition was verified, so the link stays on the 18.03 exams, which are the public ceiling you can download. Do Arnold’s problems if you buy the book. Do the exams either way.
Names and links. The problem text stays on the source site. Copyrighted contest problems are not copied here. What follows is where they are and why they hurt.
Pass: one 18.03 exam, timed, including a phase-plane problem drawn without a computer, plus a written statement of Picard–Lindelöf with the Lipschitz hypothesis actually present. Fail if “stable” means “I solved it and the formula looks small.”
Five sets. Finish a set before you touch the next. Full sentences, not a circled answer.
Ten problems to meet again. Logging a writeup schedules them at 1 day, 7 days, and 30 days in this browser.
Solve a first-order linear equation, classify an equilibrium from eigenvalues, and state Picard–Lindelöf. Out loud, no notes.