Courses / Mathematics / Differential equations: models, methods and dynamics Mathematics · unit version 1 Differential equations: models, methods and dynamics Twenty-five lessons from first-order rate models through linear systems, Laplace methods and nonlinear local analysis. Written reasoning and implementation need review; this unit alone does not establish full undergraduate ODE proficiency or examination readiness.
Open any lesson. Completion records participation; automatic practice checks and human judgment of written arguments are separate. This unit retains its own versioned requirements.
Readiness and preparation Check your starting point Try these short questions on paper, then compare your reasoning. They are optional self-checks, not a placement score or a navigation requirement. Use a targeted review when useful and return with your browser’s Back control.
Verify by differentiation that y(t)=3e^(−2t) satisfies y′=−2y and y(0)=3. Solving an equation and checking a proposed solution are distinct useful skills.
Compare reasoning and find a refresher Its derivative is −6e^(−2t), equal to −2 times y, and its value at zero is 3. Verification requires both the equation and initial condition.
Review exponential derivatives
Find ∫2t e^(t²) dt and check the result. First-order methods often require recognizing a derivative inside an integral.
Compare reasoning and find a refresher With u=t² and du=2t dt, the integral is e^(t²)+C. Differentiating recovers the original integrand.
Review substitution
For A=diag(2,−1), what happens to the coordinate vectors under A? The later systems lessons use eigenvectors to separate coupled linear motion.
Compare reasoning and find a refresher A(1,0)=2(1,0) and A(0,1)=−1(0,1). They are eigenvectors with eigenvalues 2 and −1. This recap is needed for the systems section, not to begin first-order equations.
Review eigenvalues before the systems lessons
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Differential equations: models, methods and verification
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Lesson details and objectives Separate an equation, its solution family and an initial-value problem.
Translate rate statements into equations with units. Verify a proposed solution and initial condition independently. Distinguish order, linearity and domain of a solution. 6 practice tasks.
Lesson details and objectives Construct a field and distinguish its slopes from solution values.
Evaluate direction-field slopes. Trace qualitatively consistent solution curves. Use nullclines without confusing them with solutions. 6 practice tasks.
Lesson details and objectives Use local hypotheses and separate uniqueness from global continuation.
State a sufficient local existence-uniqueness condition. Diagnose failed hypotheses without overclaiming. Distinguish local solvability from finite-time blow-up. 6 practice tasks.
Lesson details and objectives Track divisions, constants and solution intervals.
Solve separable equations with initial conditions. Preserve constant solutions lost by division. Verify implicit solutions and domains. 6 practice tasks.
Lesson details and objectives Connect growth, decay and cooling to initial data and interpretation.
Solve constant proportional-rate models. Infer a rate parameter from a ratio. Distinguish exponential approach from exact finite-time arrival. 6 practice tasks.
Lesson details and objectives Connect equilibria, rate maxima and separated solutions.
Solve logistic initial-value problems. Classify equilibria and rate behavior. State the assumptions behind a carrying-capacity model. 6 practice tasks.
Lesson details and objectives Normalize the equation and choose the integrating factor on a valid interval.
Solve y′+p(t)y=q(t). Derive the integrating-factor identity. Handle coefficient singularities and initial conditions. 6 practice tasks.
Lesson details and objectives Use inflow-minus-outflow with changing volume and consistent units.
Write a mass-balance ODE. Track volume and perfect-mixing assumptions. Check limiting concentrations and physical time domains. 6 practice tasks.
Lesson details and objectives Construct Euler steps and compare their error and stability with an exact solution.
Perform explicit Euler updates. Distinguish local and global error under stated smoothness. Analyze the decay-model stability multiplier. 6 practice tasks.
Lesson details and objectives Use the characteristic equation and two initial conditions.
Solve constant-coefficient second-order homogeneous equations with distinct real roots. Use linear superposition only for homogeneous solutions. Determine coefficients from initial data. 6 practice tasks.
Lesson details and objectives Use polynomial factors and sine/cosine pairs with real initial data.
Construct repeated-root and complex-root real solutions. Check independence and initial data. Interpret exponential envelopes separately from oscillation. 6 practice tasks.
Lesson details and objectives Use undetermined coefficients and account for overlap.
Choose suitable polynomial/exponential/trigonometric trials. Multiply trials by powers of t when resonance causes overlap. Verify the total solution against forcing and initial data. 6 practice tasks.
Lesson details and objectives Derive a general particular solution from a known fundamental pair.
Set up the two auxiliary equations. Use a nonzero Wronskian on a regular interval. Handle arbitrary forcing without hiding domain restrictions. 6 practice tasks.
Lesson details and objectives Connect damping regimes, natural frequency and forced behavior.
Classify a mass-spring equation with positive parameters. Distinguish transient, steady response and resonance. Use an energy derivative to check damping. 6 practice tasks.
Lesson details and objectives Solve diagonalizable constant systems and interpret their modes.
Form a first-order system from a higher-order equation. Use eigenvalues and eigenvectors to construct solutions. Fit vector initial conditions and check the result. 6 practice tasks.
Lesson details and objectives Handle complex and defective matrices while distinguishing local geometry from formulas.
Classify planar linear equilibria from eigenvalues. Solve a simple defective system with a t factor. Interpret centers and spirals with time orientation. 6 practice tasks.
Lesson details and objectives Track the transform domain and initial terms rather than replacing derivatives blindly.
Define the one-sided Laplace transform. Use derivative transform formulas with initial values. Invert basic rational transforms on a valid convergence half-plane. 6 practice tasks.
Lesson details and objectives Use delayed functions with correct arguments and distinguish ideal impulses from ordinary forces.
Transform unit-step shifted signals. Construct piecewise solutions by superposition. Interpret impulse jumps through integration of the equation. 6 practice tasks.
Lesson details and objectives Derive the response kernel and keep causal time limits.
Compute causal convolution integrals. Use the transform product theorem. Separate zero-input response from forced response. 6 practice tasks.
Lesson details and objectives Distinguish hyperbolic conclusions, inconclusive centers and nonlinear counterexamples.
Find equilibria and Jacobian linearizations. Apply local hyperbolic stability conclusions. Recognize limits of zero-real-part and global predictions. 6 practice tasks.
Lesson details and objectives Derive the frequency response and distinguish three frequencies with different meanings.
Derive sinusoidal response coefficients and phase with correct quadrants. Optimize displacement gain under explicit damping assumptions. Separate a sinusoidal particular from an initial-value solution. 6 practice tasks.
Lesson details and objectives Carry harmonic response into causal transforms and justify an infinite periodic synthesis.
Derive transfer and initial-state terms separately. Synthesize finite periodic inputs by linearity. Prove uniform output/derivative tail bounds for a continuous cusped input. 6 practice tasks.
Lesson details and objectives Translate a piecewise graph into slopes, signed impulses and an action on test functions.
Prove the first and second jump-derivative formulas by integration by parts. Distinguish position jumps, slope jumps, delta and delta-prime terms. Translate a finite pulse into causal convolution limits. 6 practice tasks.
Lesson details and objectives Construct a full solution operator, verify its initial normalization and carry it through a basis change.
Derive Φ(t)Φ(t₀)⁻¹ as the initial-state solution operator. Construct diagonalizable and defective exponential matrices. Prove changed-coordinate conjugacy and verify a reconstructed state. 6 practice tasks.
Lesson details and objectives Use trace, determinant and discriminant together, then resolve the exceptional cases by structure.
Partition the entire real parameter line using all relevant boundaries. Resolve zero eigenvalues, defective repeats and centers explicitly. Prove rotation direction and separate linear from nonlinear conclusions. 6 practice tasks.
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