Differential Equations & PDEs

Prerequisite path Preparation & skip guidance · Real Analysis · Linear Algebra

Studying Differential Equations & PDEs

The equation, initial data and boundary conditions together determine the problem. Lebl gives an entry through first-order methods, systems and separated heat and wave equations. Dyatlov studies distributions and elliptic regularity at graduate level. Choose the computational route to learn what solutions and boundary conditions do; choose the analytic route only when measure, functional and Fourier analysis are available as working tools.

What to work on

Solve separable and first-order linear initial-value problems.
Solve a constant-coefficient system using eigenvalues or matrix exponentials.
Derive separated solutions of the heat and wave equations with boundary conditions.
Check that a proposed solution satisfies both the equation and the initial or boundary data.

A starting text

Notes on Diffy Qs: Differential Equations for Engineers — Jiří Lebl. Read 1.3-1.4 for first-order methods, 3.3-3.4 for linear systems and 3.8 for matrix exponentials, and 4.1-4.4 for Fourier boundary expansions. Continue with 4.6-4.7 for the heat and wave equations; 4.8 supplies d'Alembert's alternative on the line.

Choosing what to read

Verify a proposed solution by substitution and check its data separately. For a linear system, compare a matrix exponential with the geometry of its phase portrait. In separation of variables, derive the boundary eigenfunctions before writing a series and explain what convergence would justify the final substitution. A formal solution formula does not establish existence or uniqueness. When you need weak derivatives, pause to learn test functions and distributions before taking on elliptic regularity.

Notes on Diffy Qs: Differential Equations for Engineers

Jiří Lebl

Connect solution techniques with initial and boundary data.

How it reads: Examples and calculations from initial-value ODEs to separated PDEs.
What it expects: Calculus and linear algebra; Fourier-series basics for boundary problems.
What to keep in mind: Separated solutions do not replace distributions, Sobolev spaces or a general PDE existence theory.
Practice: Public exercises and Solutions to Selected Exercises are present in the PDF. Answers for numbered heat-equation exercises in section 4.6 were inspected; a complete worked key is not claimed. Exercise sets in sections 1.3-1.4, 3.4, 3.8 and 4.6-4.7; compare only the matching numbered entries in Solutions to Selected Exercises. Section 4.6 answers appear on printed page 454.

Complete public PDF.

Edition, reading range and source

Read 1.3-1.4 for first-order methods, 3.3-3.4 for linear systems and 3.8 for matrix exponentials, and 4.1-4.4 for Fourier boundary expansions. Continue with 4.6-4.7 for the heat and wave equations; 4.8 supplies d'Alembert's alternative on the line.

Source · Checked 2026-10-16

Distributions, Elliptic Regularity, and Applications to PDEs

Semyon Dyatlov

Move from formal PDE manipulations to distributional and regularity arguments.

How it reads: Graduate distribution and elliptic-regularity course with geometric applications.
What it expects: Real and functional analysis, Fourier analysis and readiness for weak derivatives.
What to keep in mind: A specialized elliptic route; it is not an elementary survey of all PDE types.
Practice: The October 2, 2026 contents list Notes and exercises in sections 1.4 and 2.4; public solution coverage was not audited.

Complete 266-page MIT author PDF, dated October 2, 2026.

Edition, reading range and source

Chapter 1, sections 1.2-1.4, for function spaces, convolution and exercises; chapter 2, sections 2.1-2.4, for distribution foundations and exercises. Continue to elliptic regularity only after these.

Source · Checked 2026-10-17

Other ways into the subject

Ordinary Differential Equations and Dynamical Systems — Gerald Teschl. Sections 2.1-2.2 on fixed points and existence/uniqueness, then 2.4 and 2.6 on initial-data dependence and extensibility. Add a proof-based ODE existence course when explicit solution methods are already familiar.

Practice

Initial-value and separated-PDE exercises. Exercise sets in sections 1.3-1.4, 3.4, 3.8 and 4.6-4.7; compare only the matching numbered entries in Solutions to Selected Exercises. Section 4.6 answers appear on printed page 454.. Public exercises and Solutions to Selected Exercises are present in the PDF. Answers for numbered heat-equation exercises in section 4.6 were inspected; a complete worked key is not claimed.

Try these before moving on

Try a proof or calculation before opening the hint. These solutions address the checks below, not the exercises in the linked textbooks.

Solve y' + 2y = exp(-t), y(0)=0, and verify the result by substitution.
A hint
Multiply by exp(2t) so that the left side becomes a product derivative.
Work through the solution
Multiplication by $e^{2t}$ gives $(e^{2t}y)'=e^t$ . Integrating from $0$ to $t$ and using $y(0)=0$ yields $e^{2t}y(t)=e^t-1$ , hence $y(t)=e^{-t}-e^{-2t}$ for every real $t$ .

Its derivative is $y'=-e^{-t}+2e^{-2t}$ , so $y'+2y=e^{-t}$ , and $y(0)=1-1=0$ . For uniqueness among differentiable solutions on an interval containing $0$ , subtract two solutions. Their difference $w$ satisfies $(e^{2t}w)'=0$ and $w(0)=0$ , hence $w=0$ . This verifies both the equation and the initial-value constraint.

If this was the sticking point:
- Calculus — Review the product rule, fundamental theorem of calculus and exponential differentiation.

Already comfortable with this? Skip first-order linear and heat-equation review if you can solve and verify both checks. Assess separable equations, linear systems and waves separately through their exercise sets. Read Teschl if you can calculate solutions but cannot state a local uniqueness hypothesis or explain how a solution may cease to extend. Weak PDE solutions and Sobolev spaces require a further course.

Review the prerequisites · Compare the full subject collection

Where this can lead

Dynamical Systems & Ergodic Theory — Investigate stability and invariant sets beyond explicit ODE solutions.
Functional Analysis — Develop completeness and operator methods needed for weak and variational PDE routes.

Reading-source dates

Notes on Diffy Qs: Differential Equations for Engineers · 2026-10-16
Ordinary Differential Equations and Dynamical Systems · 2026-10-16

Reading choices revised 2026-10-17.

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From a calculated solution to a weak derivative

An explicit solution is a good place to start, but it does not explain what to do when a function has a corner. The absolute value function is enough to see why the definition of derivative changes in weak PDE methods.

What to read for this step

Keep Lebl as the starting text: §§1.3–1.4 for first-order equations, §§3.3–3.4 and §3.8 for linear systems, then §§4.1–4.4 and §§4.6–4.7 for Fourier methods, heat and waves. Verify initial and boundary data as well as the equation. The existing exercise link has selected answers, not a complete public worked key.

For this bridge, read Hunter §3.1, Definition 3.1, then §3.2, Example 3.3. It gives a short first encounter with weak derivatives. The existing Dyatlov comparison is the longer distribution-theory route; choose it when ready for test-function spaces and convolution. Teschl remains useful for ODE existence and uniqueness, not as a substitute for weak PDE methods.

Starting text · Compare reading approaches · Exercises and available answers

One example to keep in view

On $(-1,1)$ , $u(x)=|x|$ has a corner at $0$ . Its classical derivative is $-1$ on the left and $1$ on the right. The weak derivative is represented by these same values, with the value at $0$ left arbitrary. The claim concerns integrals against every smooth compactly supported test function, not a rule to ignore every exceptional point in any argument.

The two panels distinguish the graph of $u$ from a representative of its weak derivative. The holes at $0$ in the right panel leave that representative’s value unspecified; they do not leave a gap in the weak-derivative definition. Work through the check below before using $W^{1,p}$ notation.

Initial-value check, hint and solution

Absolute value and its weak derivative

The left vertex belongs to the graph. The right panel specifies

g

only away from

0

; changing

g(0)

does not change any integral or the weak derivative. The domain is

(-1,1)

, so its endpoints are open. Original schematic drawn for this example.

Try the test-function identity

On $I=(-1,1)$ let $u(x)=|x|$ . Show that its weak derivative is $g(x)=\operatorname{sgn}(x)$ for $x\ne0$ , with any value assigned at $0$ . Write the test-function identity and explain why the corner does not prevent it.

A hint

Take $\varphi\in C_c^\infty((-1,1))$ and split the integral at $0$ . On the left $|x|=-x$ ; on the right $|x|=x$ . Integrate by parts on each half. The endpoint terms vanish: $\varphi$ is zero near $\pm1$ , and the factor $x$ is zero at $0$ .

Work through the solution

Both $u$ and $g$ are locally integrable. For every $\varphi\in C_c^\infty((-1,1))$ , the required identity is

\int_{-1}^{1}|x|\varphi'(x)\,dx=-\int_{-1}^{1}g(x)\varphi(x)\,dx.

Extend $\varphi$ by zero near the endpoints. Splitting the integral gives

\begin{aligned} \int_{-1}^{1}|x|\varphi'(x)\,dx &=\int_{-1}^{0}(-x)\varphi'(x)\,dx+\int_{0}^{1}x\varphi'(x)\,dx\\ &=[-x\varphi(x)]_{-1}^{0}+\int_{-1}^{0}\varphi(x)\,dx +[x\varphi(x)]_{0}^{1}-\int_{0}^{1}\varphi(x)\,dx\\ &=\int_{-1}^{0}\varphi(x)\,dx-\int_{0}^{1}\varphi(x)\,dx =-\int_{-1}^{1}g(x)\varphi(x)\,dx. \end{aligned}

This proves the identity for every test function, not just for a few choices. The value $g(0)$ changes no Lebesgue integral: weak derivatives are determined only up to equality almost everywhere. There is still no classical derivative of $u$ at $0$ . In fact $u\in W^{1,p}((-1,1))$ for $1\le p\le\infty$ , since $u$ and $g$ are bounded on this finite interval.

If the endpoint terms were unclear, review compact support and integration by parts. If the “almost everywhere” qualification was unclear, review null sets before moving to Sobolev spaces. This calculation is an independently written example using Hunter’s definition; his Example 3.3 treats $x^+$ , not $|x|$ .

Lecture Notes on PDEs — Chapter 3: Sobolev spaces — John K. Hunter. §3.1, Definition 3.1, printed p. 47; §3.2, Example 3.3, printed p. 48 (chapter PDF pages 3–4). Free full chapter PDF on the author’s university site. Source page · Source checked 2026-10-19.

When to review, when to continue

Skip the integrating-factor warm-up only if you can verify the equation and initial value without the solution. Skip this weak-derivative example only if you can prove its test-function identity and explain the almost-everywhere qualification. Neither check replaces the measure and function-space preparation for a weak-solution course.

Editorial choice and revision

Lebl gives equations to calculate and verify. Hunter is a short bridge to Sobolev language, while Dyatlov develops the distribution framework. The point is to choose the next step, not to start three courses at once.

Added 2026-10-19. AI-assisted drafting; independent human mathematical review has not been recorded. Existing reading-source dates are unchanged.

The study guide above is translated; catalogue annotations below retain their original language.

Notes on Diffy Qs: Differential Equations for Engineers

PDF

Jiří Lebl · Textbook · Introductory

Introduces ODEs with computation, modeling and examples. A useful first encounter, but its engineering orientation should be supplemented by proof-based analysis.

Preparation: Calculus; basic linear algebra.

Open official PDF About / terms

Distributions, Elliptic Regularity, and Applications to PDEs

PDF

Semyon Dyatlov · Course notes · Advanced

Develops distributions and elliptic methods through applications including Hodge theory. These graduate notes are a specialized analytic path, not an elementary PDE survey.

Preparation: Real and functional analysis; Fourier analysis.

Open official PDF About / terms

Additional reading

Search the complete catalogue.

Introductory reading

Differential Equations and Linear Algebra: A Course for Science and Engineering

Grant B. Gustafson · Textbook · Introductory · English · PDF

The author page permits personal and educational use and links the complete textbook PDF.

An applied first course combines first-order equations, numerical methods, linear algebra, oscillations, Laplace transforms, phase planes, systems and series solutions.

Preparation: Single-variable calculus; linear algebra is developed within the text.

Use: Build computational and modelling foundations for later differential-equations courses.