Differential equations: solve, match and check the model
An answer is more than a formula returned by a solver. State the interval, initial condition and matching rule, check the equation away from a switch, and explain which qualitative conclusion survives a changed assumption.
Course topic: Differential Equations
Study route and exercise: University Reading List editorial project. How we curate · Sources checked:
Scope and responsibility
An undergraduate introduction to ordinary differential equations, with a later theory bridge. Lebl means edition 6/version 6.11 in 2026; Boyce–DiPrima–Meade means the selected 2021 twelfth-edition loose-leaf product; Teschl means the 2012 AMS book and the separately identified author-authorized preliminary file. The examples are original dimensionless paper models, not physical construction or treatment instructions. They do not establish an empirical system’s parameters or numerical software accuracy.
Choose your starting point
I need a free first-course route
Begin with Lebl’s first-order sections and his appendix when systems require linear algebra. Work selected exercises before reading supplied answers.
My syllabus prescribes the conventional survey
Use Boyce and colleagues in the assigned edition for its methods, qualitative questions and applications. Match the print format and any separately required course access.
I want to understand existence and dynamics
Use Teschl after an introductory computational pass and proof preparation. Its authorized free preliminary version is a theory bridge, not automatically the same pagination as the published book.
Compare the three books
Notes on Diffy Qs: Differential Equations for Engineers
Jiří Lebl · Edition: 6 · 2026
Free methods spine
- Choose it when
- You know basic calculus and want a manageable entry with worked explanations, exercises and later systems.
- Use another route when
- You need the exact question numbering of a different course or primarily seek an advanced dynamical-systems proof text.
- Selected reading
- Version 6.11: sections 1.1–1.4 for first-order foundations,1.6 for autonomous equations and 1.7 for Euler; later Chapter 3 for systems.
- Limits and trade-offs
- The author’s current file is May 2026. Answers are supplied only for a subset, not every exercise. A revision date and the edition number are different pieces of bibliographic information.
Try before choosing
For the first interval below, derive y=2−e^(−2t) with an integrating factor and verify it without a solver.
Check your work: Check both y(0)=1 and y′+2y=4. A correct antiderivative with the wrong integration constant is not the specified initial-value solution.
Elementary Differential Equations and Boundary Value Problems
William E. Boyce; Richard C. DiPrima; Douglas B. Meade · Edition: 12 · 2021
Assigned broad applied course
- Choose it when
- Your class uses the twelfth edition or you want a broad methods-and-applications reference.
- Use another route when
- You need a complete free book or only a short first-order route.
- Selected reading
- Sections 2.1 and 2.4 on first-order linear equations and linear/nonlinear distinctions;2.7 for Euler,2.8 for existence/uniqueness; selected later methods as assigned.
- Limits and trade-offs
- ISBN 9781119777694 identifies the 2021 loose-leaf text. Wiley lists separate electronic and access bundles. A print purchase alone does not establish the platform rights needed by a class.
Try before choosing
Sketch the piecewise input, then predict continuity of y and the one-sided slopes at the switching time before calculating.
Check your work: A finite step in the forcing changes the derivative, not automatically the state. State what solution concept applies at the switch rather than forcing a nonexistent two-sided derivative.
Ordinary Differential Equations and Dynamical Systems
Gerald Teschl · 2012
Free classical-theory bridge
- Choose it when
- You can follow proofs and want existence, uniqueness, dependence and dynamics behind solution methods.
- Use another route when
- You are still struggling with basic integration or want only a table of algorithms.
- Selected reading
- Chapters 1–2 for first-order qualitative ideas and initial-value theory;3 for linear systems after matrix preparation.
- Limits and trade-offs
- The author’s PDF is a preliminary version made available with AMS permission and explicit restrictions on changing/reposting. Free reading is not permission to distribute an edited copy.
Try before choosing
Explain which hypotheses you would check before applying a smooth initial-value theorem on each interval, and why the switched problem needs matching.
Check your work: Use the actual coefficient/input regularity on each open interval. A theorem with continuous forcing cannot simply be quoted across a forcing discontinuity without explaining the integral or piecewise interpretation.
Check your preparation
What calculus is needed?
Differentiate exponentials, integrate a linear equation with an integrating factor, and distinguish a function from its derivative. Systems additionally need matrix algebra; proof-oriented theory needs more mathematical maturity.
Is an initial condition part of the model?
Yes. A family of solutions is not the requested trajectory. State where the condition is imposed and whether the state is continuous across an event.
What can a direction field show?
It helps visualize slopes and possible behaviour on the stated domain. A plotted curve alone does not prove uniqueness, long-time behaviour or a numerical error bound.
Six steps with a concrete output
Outputs marked ready: 0 / 6
These are your own study marks, not an assessment. They are not sent to this site.
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Write the model before solving
Selected reading: Lebl 1.1–1.4 or Boyce 2.1.
Make this: Record the equation, initial value, dimensionless domain and two input intervals.
Check your work: Use a piecewise definition; a single expression that silently applies 4 after the switch is a different problem.
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Solve and substitute
Selected reading: The same linear-equation route.
Make this: Derive the first-interval solution with an arbitrary constant, fix it, then compute the residual y′+2y−4.
Check your work: The residual vanishes and the initial value is satisfied. Checking only one time value is insufficient.
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Match at the switch
Selected reading: Revisit first-order modeling and the stated assumptions.
Make this: Calculate the left limit at 1, impose continuity, and solve the second interval from that value.
Check your work: Keep the time shift t−1 explicit and distinguish left/right derivatives. The two interval formulas give the same state at 1.
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Check qualitative behaviour
Selected reading: Lebl 1.6 or Teschl 1.3–1.5.
Make this: Sketch monotonicity and the relevant interval equilibria, explaining the slope change at 1.
Check your work: The first segment rises toward 2; after the switch the positive state decays toward 0. A switch changes which equilibrium is relevant.
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Assess a numerical claim
Selected reading: Lebl 1.7 or Boyce 2.7; Teschl 2.7 as a later theory route.
Make this: Describe how a time step crossing 1 could use the wrong forcing. Propose splitting the mesh at the known event.
Check your work: This is a reproducibility specification, not a proved error bound. Report step size and how the switch is handled before comparing a numerical output with the analytic solution.
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Change the matching rule
Selected reading: Use the original derivation and the follow-up’s explicit update.
Make this: Solve the reset variant and explain why the new discontinuity is an added model assumption.
Check your work: A jump condition is stated, not inferred from the old bounded input step. Check the new one-sided values and the equation on each open interval.
Worked exercise
Original example: an input step changes the slope
For dimensionless t≥0, solve y′+2y=f(t), y(0)=1, where f(t)=4 for 0≤t<1 and f(t)=0 for t≥1. Require y to remain continuous at 1. Give interval formulas, y(1), one-sided slopes and limiting behaviour.
Assumptions
All values and the state are invented and exact. The solution is continuous and piecewise differentiable, with the ODE required on the two open intervals. At the forcing discontinuity use the equivalent integral/piecewise interpretation, not an asserted classical derivative at that point. There is no instantaneous state update in this first model.
- For 0≤t<1, an integrating factor gives y=2+Ce^(−2t); y(0)=1 makes C=−1. Thus y=2−e^(−2t), and y(1)=2−e⁻² by continuity.
- For t>1, the equation is homogeneous. Matching the state gives y=(2−e⁻²)e^(−2(t−1)). It approaches 0 as t grows.
- The left slope is 2e⁻² and the right slope is −2(2−e⁻²)=−4+2e⁻². Their difference is −4, matching the drop in f. The state is continuous although its derivative is not.
Answer and interpretation
Use y=2−e^(−2t) before 1 and y=(2−e⁻²)e^(−2(t−1)) after 1, with common value 2−e⁻². The trajectory rises before the switch and decays afterwards. A global continuously differentiable formula would misdescribe this stipulated model.
Test the result
Substitute each branch into its own equation and integrate over a shrinking interval around 1: a bounded forcing has vanishing area there, consistent with no state jump. The value assigned to f at the single point does not change this integral solution.
Change an assumption
Add a state update instead of assuming continuity
Keep the same two interval equations and initial value, but replace continuity by the explicit event rule y(1+)=y(1−)+3. Find the new post-event trajectory and compare it with the first model. The update is a stipulated mathematical rule, not inferred from the forcing step.
- The first interval is unchanged, so y(1−)=2−e⁻². Applying the new rule gives y(1+)=5−e⁻².
- For t>1, solve the same homogeneous equation with that right-hand value: y=(5−e⁻²)e^(−2(t−1)). Relative to the original, the difference is 3e^(−2(t−1)).
- Both post-event branches eventually tend to 0, but their transient values differ. The new trajectory is not a continuous integral solution of the old bounded-forcing equation across 1; the update must remain part of the model statement.
Answer and interpretation
The new post-event coefficient is 5−e⁻² instead of 2−e⁻². Equal long-run limits do not make the two trajectories identical or justify discarding an event rule.
Test the result
Verify the jump is exactly 3 and the post-event residual y′+2y is zero. Replacing 3 by 0 recovers the first model. A numerical scheme that merely changes f at 1 without implementing the update would solve the wrong problem.
Common mistakes
One formula for all intervals
Apply each equation on its own domain and explicitly connect the state.
Continuity and differentiability are different
A continuous state can have different one-sided derivatives. A finite forcing step is not itself a prescribed state jump.
A smooth picture is not verification
Show a residual, initial/matching checks and numerical settings; a plot may conceal the event.
Before moving on
- Derive both branches and verify the initial condition and matching rule.
- Explain which equation is required at and away from a switch.
- Distinguish equal long-run limits from equal transient solutions.
Sources and edition checks
Sources checked:
- Lebl: official version, prerequisites and contents
The author confirms the first-course audience, current 6.11 revision and selected first-order sections; answer availability is limited to specified exercises.
- Wiley: twelfth-edition loose-leaf identity and contents
Direct publisher metadata and contents confirm 2021, twelfth edition and the selected loose-leaf ISBN; digital/access products are separate.
- Teschl: author-authorized preliminary text and contents
The authorized preliminary PDF supplies the theoretical chapter map and explicitly distinguishes permission to read from permission to modify/repost.
These are proposed study activities, not official course assignments or a claim that every book has been read cover to cover.