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| | + | ==Course description== |
| | + | Partial differential equations arise in many different areas as one tries |
| | + | to describe the behavior of a system ruled by some law. Typically, this has to do with |
| | + | some physical process such as heat diffusion in a material, vibrations of a bridge, circulation |
| | + | of fluids, the behavior of microscopic particles or the evolution of the universe as a whole. |
| | + | Modeling by means of partial differential equations has been successful in other disciplines |
| | + | as well, like in the case of the Black-Scholes equation for stock options pricing and the |
| | + | Hodgkin–Huxley equations for firing patterns of a neuron. Partial differential equations are |
| | + | an important tool in Applied Math and Pure Math. This course gives an introduction to PDE's in the setting of two independent variables. |
| | + | |
| | + | '''Textbooks: |
| | + | ''' |
| | + | * P. Olver: Introduction to Partial Differential Equations (Undergraduate Texts in Mathematics) 1st ed. 2014, Corr. 3rd printing 2016 |
| | + | * L.C. Evans: Partial Differential Equations: Second Edition (Graduate Studies in Mathematics) 2nd Edition |
| | + | |
| | ==Topics List== | | ==Topics List== |
| | {| class="wikitable sortable" | | {| class="wikitable sortable" |
| | ! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes | | ! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes |
| | |- | | |- |
| − | |Week I | + | |Week 1 |
| | || | | || |
| − | * Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3 | + | * |
| | || | | || |
| − | * [[Order of Differential Equations]]
| + | Introduction and classification of PDE, Calculus review |
| | || | | || |
| − | * Integration techniques
| + | Multivariable Calculus, Chain Rule |
| − | :- [[Direct Integration]]
| |
| − | :- [[Integration by Substitution]]
| |
| − | :- [[Integration by Parts]]
| |
| − | :- [[Partial Fractions]]
| |
| | || | | || |
| − | * Explain the basic notion of the order of a differential equation. | + | * Definition of a PDE as a relation between partial derivatives of an unknown function. Classification of PDE according to order - linear/nonlinear/quasilinear |
| | |- | | |- |
| − | |Week I | + | |Week 2 |
| | || | | || |
| − | * Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3 | + | * |
| | || | | || |
| − | * [[Solutions of Differential Equations]]
| + | Applied examples of PDE |
| | || | | || |
| − | * Integration techniques
| + | Multivariable Calculus, Chain Rule |
| − | :- [[Direct Integration]]
| |
| − | :- [[Integration by Substitution]]
| |
| − | :- [[Integration by Parts]]
| |
| − | :- [[Partial Fractions]]
| |
| | || | | || |
| − | * Explain the basic notion of solutions of differential equations. | + | * Origin and background of common PDE's: heat equation, wave equation, transport equation, etc. |
| | |- | | |- |
| − | |Week I | + | |Week 3 |
| | || | | || |
| − | * Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3 | + | * |
| | || | | || |
| − | * [[Initial Value Problem|Initial Value Problem (IVP)]]
| + | The method of characteristics for first-order quasilinear equations |
| | || | | || |
| − | * Integration techniques
| + | Multivariable Calculus, Chain Rule |
| − | :- [[Direct Integration]]
| |
| − | :- [[Integration by Substitution]]
| |
| − | :- [[Integration by Parts]]
| |
| − | :- [[Partial Fractions]]
| |
| | || | | || |
| − | * Explain the basic notion of the initial values problem. | + | * Solving quasilinear first-order equations using the method of characteristics |
| | |- | | |- |
| − | |Week I | + | |Week 4 |
| | || | | || |
| − | * Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3 | + | * |
| | || | | || |
| − | * [[Cauchy Problem]]
| + | The method of characteristics for first-order fully nonlinear equations |
| | || | | || |
| − | * Integration techniques
| + | Multivariable Calculus, Chain Rule |
| − | :- [[Direct Integration]]
| |
| − | :- [[Integration by Substitution]]
| |
| − | :- [[Integration by Parts]]
| |
| − | :- [[Partial Fractions]]
| |
| | || | | || |
| − | * Explain the Cauchy Problem | + | * Solving fully nonlinear first-order equations (e.g. the Eikonal equation) using the method of characteristics |
| − | * Explain the basic notion of existence and uniqueness of a solution to the Cauchy Problem.
| |
| | |- | | |- |
| − | |Week I | + | |Week 5 |
| | || | | || |
| − | * Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3 | + | * |
| | || | | || |
| − | * [[Separation of Variables (1st Order)]]
| + | Heat and wave equation on the whole real line |
| | || | | || |
| − | * Integration techniques
| + | Differentiation of integrals with respect to a parameter, integration by parts |
| − | :- [[Direct Integration]]
| |
| − | :- [[Integration by Substitution]]
| |
| − | :- [[Integration by Parts]]
| |
| − | :- [[Partial Fractions]]
| |
| | || | | || |
| − | * Determine separable differential equations of the first order. | + | * Fundamental solution of the heat equation, D'Alembert's formula for the wave equation |
| − | * Apply direct methods to evaluate exact solutions of separable differential equations of the first order.
| |
| | |- | | |- |
| − | |Week II | + | |Week 6 |
| | || | | || |
| − | * Ahmad and Ambrosetti 2014, Chaps. 1 and 3 | + | * |
| | || | | || |
| − | * [[Homogeneous Differential Equations|Homogeneous Differential Equations (1st Order)]]
| + | Initial-boundary value problem for heat and wave equation I |
| | || | | || |
| − | * Integration techniques
| + | Partial derivatives, chain rule |
| − | :- [[Direct Integration]]
| |
| − | :- [[Integration by Substitution]]
| |
| − | :- [[Integration by Parts]]
| |
| − | :- [[Partial Fractions]]
| |
| | || | | || |
| − | * Determine homogeneous differential equations of the first order. | + | * Separation of variables method for heat and wave equation |
| − | * Apply direct methods to evaluate exact solutions of homogeneous differential equations of the first order (substitutions).
| |
| − | * Use some differential equations as mathematical models in biology, population dynamics, mechanics and electrical circuit theory problems.
| |
| | |- | | |- |
| − | |Week II | + | |Week 7 |
| | || | | || |
| − | * Ahmad and Ambrosetti 2014, Chaps. 1 and 3 | + | * |
| | || | | || |
| − | * [[Linear Differential Equations|Linear Differential Equations (1st Order)]]
| + | Initial-boundary value problem for heat and wave equation II, introduction to Fourier series |
| | || | | || |
| − | * Integration techniques
| + | Partial derivatives, chain rule |
| − | :- [[Direct Integration]]
| |
| − | :- [[Integration by Substitution]]
| |
| − | :- [[Integration by Parts]]
| |
| − | :- [[Partial Fractions]]
| |
| | || | | || |
| − | * Determine linear differential equations of the first order. | + | * Forming more general solutions out of infinite superposition of basic solutions |
| − | * Use some differential equations as mathematical models in biology, population dynamics, mechanics and electrical circuit theory problems.
| |
| | |- | | |- |
| − | |Week II | + | |Week 8 |
| | || | | || |
| − | * Ahmad and Ambrosetti 2014, Chaps. 1 and 3 | + | * |
| | || | | || |
| − | * [[Integrating Factor]]
| + | Introduction to Fourier series |
| | || | | || |
| − | * [[Linear Differential Equations|Linear Differential Equations (1st Order)]]
| + | Infinite series |
| | || | | || |
| − | * Apply integrating factor to solve linear differential equations of the first order. | + | * Orthonormal systems of functions, spectral method for the wave and heat equation |
| − | * Use some differential equations as mathematical models in biology, population dynamics, mechanics and electrical circuit theory problems.
| |
| | |- | | |- |
| − | |Week III | + | |Week 9 |
| | || | | || |
| − | * Ahmad and Ambrosetti 2014, Ch. 3 | + | * |
| | || | | || |
| − | * [[Bernoulli Equations (1st Order)]]
| + | Schroedinger equation |
| | || | | || |
| − | * [[Linear Differential Equations|Linear Differential Equations (1st Order)]]
| + | Complex numbers |
| | || | | || |
| − | * Determine Bernoulli of the first order. | + | * Basic properties of Schroedinger equation, particle in a potential well |
| − | * Apply direct methods to evaluate exact solutions of Bernoulli of the first order.
| |
| | |- | | |- |
| − | |Week III | + | |Week 10 |
| | || | | || |
| − | * Ahmad and Ambrosetti 2014, Ch. 3
| + | |
| | || | | || |
| − | * [[Exact Differential Equations|Exact Differential Equations (1st Order)]]
| + | Qualitative properties of PDE's |
| | || | | || |
| − | * [[Integrating Factor]] for exact equations.
| + | Differentiation of integrals with respect to parameter |
| − | * [[Partial Derivatives]]
| |
| | || | | || |
| − | * Determine Exact Differential Equations of the first order. | + | * Uniqueness of solutions, finite and infinite propagation speed for wave and heat equation |
| − | * Apply direct methods to evaluate exact solutions of Exact Differential Equations of the first order.
| |
| − | * Use the integrating factor technique for exact equations.
| |
| | |- | | |- |
| − | |Week IV | + | |Week 11 |
| | || | | || |
| − | * Ahmad and Ambrosetti 2014, Chaps. 1-3 | + | * |
| | || | | || |
| − | * Overview of the solutions methods discussed so far (Chapters 1-3).
| + | Introduction to numerical methods for PDE (optional) |
| | || | | || |
| − | * Integration techniques
| + | Derivatives, Calculus, Matrices, Linear Algebra |
| − | :- [[Direct Integration]]
| |
| − | :- [[Integration by Substitution]]
| |
| − | :- [[Integration by Parts]]
| |
| − | :- [[Partial Fractions]]
| |
| − | * [[Partial Derivatives]]
| |
| − | * First-order differential equations:
| |
| − | :- [[Separation of Variables (1st Order)]]
| |
| − | :- [[Homogeneous Differential Equations|Homogeneous Differential Equations (1st Order)]]
| |
| − | :- [[Linear Differential Equations|Linear Differential Equations (1st Order)]]
| |
| − | :- [[Bernoulli Equations (1st Order)]]
| |
| − | :- [[Exact Differential Equations|Exact Differential Equations (1st Order)]]
| |
| | || | | || |
| − | * Determine the type of different classes of differential equations of the first order: separable, linear, homogeneous, Bernoulli, exact. | + | * Basic finite difference schemes for first-order quasilinear equations, CFL condition |
| − | * Use direct methods to solve first order differential equations solved and not solved for the first derivative.
| |
| | |- | | |- |
| − | |Week V | + | |Week 12 |
| | || | | || |
| − | * Ahmad and Ambrosetti 2014, Ch. 5
| + | |
| | || | | || |
| − | * [[Linear Independence of Functions]].
| + | Introduction to the Laplace and Poisson equation |
| | || | | || |
| − | * [[Linear Independence of Vectors]]. | + | * |
| | || | | || |
| − | * Understanding of Linear Independence of Functions. | + | * Solving the Laplace equation on the whole space and on a simple bounded region (square, disc) |
| | |- | | |- |
| − | |Week V | + | |Week 13 |
| | || | | || |
| − | * Ahmad and Ambrosetti 2014, Ch. 5 | + | * |
| | || | | || |
| − | * [[Linear Independence of Functions|Linear Dependence of Functions]].
| + | Introduction to the Calculus of Variations |
| | || | | || |
| − | * [[Linear Dependence of Vectors]].
| + | Differentiation of an integral with respect to a parameter, parametric surfaces |
| | || | | || |
| − | * Understanding of Linear Dependence of Functions. | + | * Compute the variational derivative of a functional |
| | |- | | |- |
| − | |Week V | + | |Week 14 |
| | || | | || |
| − | * Ahmad and Ambrosetti 2014, Ch. 5 | + | * |
| | || | | || |
| − | * [[Wronskian]]
| + | Review, advanced topics |
| | || | | || |
| − | * [[Linear Independence of Functions]]. | + | * |
| − | * [[Linear Independence of Functions|Linear Dependence of Functions]].
| |
| − | * [[Determinant]].
| |
| | || | | || |
| − | * Showing linear independence of two functions using the Wronskian. | + | * |
| − | * Showing linear independence of two solutions of Linear Second-Order ODEs using the Wronskian.
| |
| − | |-
| |
| − | |Week VI
| |
| − | ||
| |
| − | * Ahmad and Ambrosetti 2014, Ch. 5
| |
| − | ||
| |
| − | * [[Reduction of the Order]]
| |
| − | ||
| |
| − | * [[Wronskian]].
| |
| − | * [[Quadratic Equations]].
| |
| − | * [[Linear Differential Equations|Linear Differential Equations (1st Order)]].
| |
| − | * [[Solutions of Linear Systems]].
| |
| − | ||
| |
| − | * Apply of the reduction of the order technique for second-order ODEs with a given solution.
| |
| − | |-
| |
| − | |Week VI
| |
| − | ||
| |
| − | * Ahmad and Ambrosetti 2014, Ch. 5
| |
| − | ||
| |
| − | * [[Homogeneous Differential Equations|Linear Homogeneous Equations]]
| |
| − | ||
| |
| − | * [[Wronskian]].
| |
| − | * [[Quadratic Equations]].
| |
| − | * [[Linear Differential Equations|Linear Differential Equations (1st Order)]].
| |
| − | * [[Solutions of Linear Systems]].
| |
| − | ||
| |
| − | * Determine homogeneous classes of differential equations of the second and higher order.
| |
| − | * Determine linear and non-linear classes of differential equations of the second and higher order.
| |
| − | |-
| |
| − | |Week VI
| |
| − | ||
| |
| − | * Ahmad and Ambrosetti 2014, Ch. 5
| |
| − | ||
| |
| − | * [[Abel’s Theorem]]
| |
| − | ||
| |
| − | * [[Wronskian]].
| |
| − | * [[Quadratic Equations]].
| |
| − | * [[Linear Differential Equations|Linear Differential Equations (1st Order)]].
| |
| − | * [[Solutions of Linear Systems]].
| |
| − | ||
| |
| − | * Determine Wronskian for a second-order ODE with 2 given solutions.
| |
| − | |-
| |
| − | |Week VI
| |
| − | ||
| |
| − | * Ahmad and Ambrosetti 2014, Ch. 5
| |
| − | ||
| |
| − | * [[Fundamental Solutions]]
| |
| − | ||
| |
| − | * [[Wronskian]].
| |
| − | * [[Quadratic Equations]].
| |
| − | * [[Linear Differential Equations|Linear Differential Equations (1st Order)]].
| |
| − | * [[Solutions of Linear Systems]].
| |
| − | ||
| |
| − | * Determine fundamental solutions.
| |
| − | |-
| |
| − | |Week VI
| |
| − | ||
| |
| − | * Ahmad and Ambrosetti 2014, Ch. 5
| |
| − | ||
| |
| − | * [[Linear Differential Equations|Linear Non-homogeneous Equations]]
| |
| − | ||
| |
| − | * [[Wronskian]].
| |
| − | * [[Quadratic Equations]].
| |
| − | * [[Linear Differential Equations|Linear Differential Equations (1st Order)]].
| |
| − | * [[Solutions of Linear Systems]].
| |
| − | ||
| |
| − | * Determine non-homogeneous classes of differential equations of the second and higher order.
| |
| − | * Determine linear and non-linear classes of differential equations of the second and higher order
| |
| − | |-
| |
| − | |Week VI
| |
| − | ||
| |
| − | * Ahmad and Ambrosetti 2014, Ch. 5
| |
| − | ||
| |
| − | * [[Variation Of Parameters (2nd Order)|Variation of Parameters (2nd Order)]]
| |
| − | ||
| |
| − | * Integration techniques
| |
| − | :- [[Direct Integration]]
| |
| − | :- [[Integration by Substitution]]
| |
| − | :- [[Integration by Parts]]
| |
| − | :- [[Partial Fractions]]
| |
| − | * [[Quadratic Equations]].
| |
| − | * [[Solutions of Linear Systems]].
| |
| − | ||
| |
| − | * Apply of the variation of parameters technique for second-order ODEs.
| |
| − | |-
| |
| − | |Week VII
| |
| − | ||
| |
| − | * Ahmad and Ambrosetti 2014, Ch. 5
| |
| − | ||
| |
| − | * [[Variation Of Parameters (2nd Order)|Variation of Parameters (2nd Order)]] (continued)
| |
| − | ||
| |
| − | * Integration techniques
| |
| − | :- [[Direct Integration]]
| |
| − | :- [[Integration by Substitution]]
| |
| − | :- [[Integration by Parts]]
| |
| − | :- [[Partial Fractions]]
| |
| − | * [[Quadratic Equations]].
| |
| − | * [[Solutions of Linear Systems]].
| |
| − | ||
| |
| − | * Apply variation of parameters technique for second-order ODEs.
| |
| − | |-
| |
| − | |Week VII
| |
| − | ||
| |
| − | * Ahmad and Ambrosetti 2014, Ch. 5
| |
| − | ||
| |
| − | * [[Method of Undetermined Coefficients (2nd Order)]]
| |
| − | ||
| |
| − | * [[Quadratic Equations]].
| |
| − | * [[Systems of Linear Equations]].
| |
| − | ||
| |
| − | * Apply method of undetermined coefficients technique for second-order ODEs.
| |
| − | |-
| |
| − | |Week VIII
| |
| − | ||
| |
| − | * Ahmad and Ambrosetti 2014, Ch. 5
| |
| − | ||
| |
| − | * [[Non-linear 2nd Order ODEs]]
| |
| − | ||
| |
| − | * [[Algebraic Equations]]
| |
| − | * [[Reduction of the Order]]
| |
| − | * Integration techniques
| |
| − | :- [[Direct Integration]]
| |
| − | :- [[Integration by Substitution]]
| |
| − | :- [[Integration by Parts]]
| |
| − | :- [[Partial Fractions]]
| |
| − | ||
| |
| − | * Methods for nonlinear second-order ODEs.
| |
| − | * Apply reduction of the order method to some nonlinear second-order ODEs.
| |
| − | |-
| |
| − | |Week VIII
| |
| − | ||
| |
| − | * Ahmad and Ambrosetti 2014, Ch. 5
| |
| − | ||
| |
| − | * [[Variation Of Parameters|Variation of Parameters (Higher Order)]]
| |
| − | ||
| |
| − | * [[Variation Of Parameters|Variation of Parameters (2nd Order)]]
| |
| − | ||
| |
| − | * Apply variation of parameters technique for higher-order ODEs
| |
| − | |-
| |
| − | |Week VIII
| |
| − | ||
| |
| − | * Ahmad and Ambrosetti 2014, Ch. 5
| |
| − | ||
| |
| − | * [[Method of Undetermined Coefficients|Method of Undetermined Coefficients (Higher Order)]]
| |
| − | ||
| |
| − | * [[Method of Undetermined Coefficients (2nd Order)]]
| |
| − | ||
| |
| − | * Apply method of undetermined coefficients technique for higher-order ODEs
| |
| − | |-
| |
| − | |Week IX
| |
| − | ||
| |
| − | * Ahmad and Ambrosetti 2014, Ch. 6
| |
| − | ||
| |
| − | * [[Linear Differential Equations|Linear Differential Equations (Higher Order)]]
| |
| − | ||
| |
| − | * [[Linear Differential Equations|Linear Differential Equations (1st Order)]]
| |
| − | * [[Variation Of Parameters|Variation of Parameters (Higher Order)]].
| |
| − | * [[Method of Undetermined Coefficients|Method of Undetermined Coefficients (Higher Order)]].
| |
| − | ||
| |
| − | * Methods for linear higher-order ODEs
| |
| − | |-
| |
| − | |Week X
| |
| − | ||
| |
| − | * Ahmad and Ambrosetti 2014, Chaps. 5, 6
| |
| − | ||
| |
| − | * Overview of the solutions methods for second and higher order differential equations.
| |
| − | ||
| |
| − | * [[Algebraic Equations]]
| |
| − | * Direct methods for second and higher-order ODEs:
| |
| − | :- [[Variation Of Parameters|Variation of Parameters (Higher Order)]]
| |
| − | :- [[Method of Undetermined Coefficients|Method of Undetermined Coefficients (Higher Order)]]
| |
| − | ||
| |
| − | * Evaluate the exact solutions of important classes of differential equations such as second order differential equations as well as some higher order differential equations.
| |
| − | |-
| |
| − | |Week X
| |
| − | ||
| |
| − | * Ahmad and Ambrosetti 2014, Chaps. 10
| |
| − | ||
| |
| − | * [[Power Series Solutions]]
| |
| − | ||
| |
| − | * [[Power Series Induction]]
| |
| − | ||
| |
| − | Apply power series method to evaluate solutions of first-order and second-order ODEs.
| |
| − | |-
| |
| − | |Week XI
| |
| − | ||
| |
| − | * Ahmad and Ambrosetti 2014, Chaps. 10
| |
| − | ||
| |
| − | * [[Power Series Solutions]] (continued)
| |
| − | ||
| |
| − | * [[Power Series Induction]]
| |
| − | ||
| |
| − | Apply power series method to evaluate solutions of first-order and second-order ODEs.
| |
| − | |-
| |
| − | |Week XII
| |
| − | ||
| |
| − | * Ahmad and Ambrosetti 2014, Ch. 11
| |
| − | ||
| |
| − | * [[Laplace Transform]]
| |
| − | ||
| |
| − | * [[Functions]] of Single Variable.
| |
| − | * [[Continuity]] of functions of single variables.
| |
| − | * [[Derivatives]] of functions of single variables.
| |
| − | * [[Improper Integrals]] of functions of single variables with infinite limits.
| |
| − | ||
| |
| − | * Definition and main properties of the L-transform.
| |
| − | |-
| |
| − | |Week XIII
| |
| − | ||
| |
| − | * Ahmad and Ambrosetti 2014, Ch. 11
| |
| − | ||
| |
| − | * [[Inverse Laplace Transform]]
| |
| − | ||
| |
| − | * [[Laplace Transform]]
| |
| − | * [[Complex Derivatives]]
| |
| − | ||
| |
| − | * Apply the theorem(s) for inverse L-transform.
| |
| − | |-
| |
| − | |Week XIV
| |
| − | ||
| |
| − | * Ahmad and Ambrosetti 2014, Ch. 11
| |
| − | ||
| |
| − | * [[Laplace Transform to ODEs]]
| |
| − | ||
| |
| − | * [[Linear Differential Equations|Linear Equations]]
| |
| − | * [[Laplace Transform]]
| |
| − | * [[Inverse Laplace Transform]]
| |
| − | ||
| |
| − | * Apply the Laplace transform as solution technique.
| |
| − | |-
| |
| − | |Week XIV
| |
| − | ||
| |
| − | * Ahmad and Ambrosetti 2014, Ch. 11
| |
| − | ||
| |
| − | * [[Laplace Transform to ODEs|Laplace Transform to Systems of ODEs]]
| |
| − | ||
| |
| − | * [[Solutions of Linear Systems]].
| |
| − | * [[Laplace Transform]].
| |
| − | * [[Inverse Laplace Transform]].
| |
| − | ||
| |
| − | * Apply the Laplace transform as solution technique.
| |
| − | |-
| |
| − | |Week XV
| |
| − | ||
| |
| − | * Ahmad and Ambrosetti 2014
| |
| − | ||
| |
| − | * Overview of the solutions methods discussed.
| |
| − | ||
| |
| − | * [[Separation of Variables (1st Order)]]
| |
| − | * [[Homogeneous Differential Equations|Homogeneous Differential Equations (1st Order)]]
| |
| − | * [[Linear Differential Equations|Linear Differential Equations (1st Order)]]
| |
| − | * [[Integrating Factor]]
| |
| − | * [[Bernoulli Equations (1st Order)]]
| |
| − | * [[Exact Differential Equations|Exact Differential Equations (1st Order)]]
| |
| − | * [[Reduction of the Order]]
| |
| − | * [[Method of Undetermined Coefficients (2nd Order)]]
| |
| − | * [[Non-linear 2nd Order ODEs]]
| |
| − | * [[Variation Of Parameters|Variation of Parameters (Higher Order)]]
| |
| − | * [[Method of Undetermined Coefficients|Method of Undetermined Coefficients (Higher Order)]]
| |
| − | * [[Linear Differential Equations|Linear Differential Equations (Higher Order)]]
| |
| − | * [[Power Series Solutions]]
| |
| − | * [[Laplace Transform to ODEs]]
| |
| − | * [[Laplace Transform to ODEs|Laplace Transform to Systems of ODEs]]
| |
| − | ||
| |
| − | * Apply all solutions methods discussed.
| |
| | |} | | |} |
