SYLLABUS of the course MATHEMATICAL METHODS FOR EXPERIMENTAL SCIENCE

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1 SYLLABUS of the course MATHEMATICAL METHODS FOR EXPERIMENTAL SCIENCE Bachelor in Computer Science and Engineering, University of Bolzano-Bozen, a.y Lecturer: LEONARDO RICCI (last updated on January 18, 2018) 1

2 (02/10/2017) 1. Introduction to the course. Vectors (part 1 of 2). Vectors (part 1 of 2): vectors; n-space; scalars; properties of vectors; graphical representation; scalar product and its properties; length or norm of a vector; Cauchy-Schwartz inequality; triangle inequality (derived from Cauchy-Schwartz inequality). (02/10/2017) 2. Vectors (part 2 of 2). A summary of single-variable differentiation. Vectors (part 2 of 2): perpendicularity or orthogonality (case on R 2 and generalization to R n ); projection of a vector onto another (case on R 2 and generalization to R n ); angle between two vectors (case on R 2 and generalization to R n ); unit vectors (versors); standard basis versors; representation theorem; graphical representation. A summary of single-variable differentiation: definition and graphical interpretation; derivative of f(x) = x and f(x) = x 2 ; derivative of the product of two functions. (03/10/2017) E1. Exercise class on vectors, and on single-variable differentiation. 2

3 Solution of exercises on vectors: proof of x+y 2 x 2 +y 2 2 via Cauchy Schwartz inequality; generalization of the result (the modulus of the average is the root mean square) to an arbitrary number of variables; projection of A = (1, 2, 3) along B = (1, 2, 2); projection of A = (1, 2, 3, 4) along B = (1, 1, 1, 1); expression for cos(x y) by using the angle formed by two vectors; a summary of trigonometric formulas. Solution of exercises on single-variable differentiation: derivative of f(x) = x n, n N; differentiation of a linear combination of functions; exercise, derivative of f(x) = x x2 +2; derivative of f(x) = sin(x) and f(x) = cos(x). (09/10/2017) 3. Functions of several variables: definitions, graphs, level curves. A summary of single-variable differentiation. From the functions of a single variable to the functions of several variables: definition; domain; range. Examples concerning domains and ranges from the textbook: paragraph 12.1, exercises 1, 2, p Graphs (expecially of functions of two variables): definition; graphical visualization of 2-d case (artist s view); graph as an n-dimensional hypersurface in an n+1 dimensional space. Level curves (expecially of functions of two variables): definition; examples, 3

4 contour lines, or contours (elevation), isobars and isotherms; extension to the 3-d case (level surfaces). Examples of graphs and level curves: z = 1 x 2 y 2 (via gnuplot); z = xy (via gnuplot); ( ) z = sin x 2 +y 2 (via gnuplot). x 2 +y 2 A summary of single-variable differentiation: chain rule; example of f(x) = sin(x 2 ). (10/10/2017) 4. Functions of several variables: limits and continuity. A summary of single-variable differentiation. A summary on limits in the single variable (1-d) case: summary, by using the example lim x 3 x 2 = 9; other examples, lim x 0 x x, lim x 0+ x, lim x 0 sinx x. Limits in the n-d case: definition; existence of the limit for every path chosen to reach the limit point, and independence on the path; extension of usual 1-d laws of limits (addition, subtraction, multiplication, division, composition) to n-d cases; extension of the 1-d squeeze theorem to n-d cases (statement only). Examples of limits from the textbook: paragraph 12.2, example 3, p. 678, lim (x,y) (0,0) 2xy x 2 +y 2 ; paragraph 12.2, example 4, p. 679, lim (x,y) (0,0) 2x 2 y x 4 +y 2 ; paragraph 12.2, example 5, p. 679, lim (x,y) (0,0) x 2 y x 2 +y 2. Continuity: 4

5 summary of the single variable (1-d) case; definition; extension of usual 1-d laws (sum, difference, product, quotient, composition) to n-d cases. Continuous extension to a point: continuous extension to a point in 1-d (ex.gr.: extension of y = sinx x in 0); continuous extension to a point (ex.gr.: extension of z = x2 y x 2 +y 2 in 0; see also paragraph 12.2, example 5, p. 679). A summary of single-variable differentiation: derivative of a quotient; example of sinx x. (10/10/2017) E2. Exercise class on domains and limits of functions of several variables, and on single-variable differentiation. Solution of selected exercises on domains of functions of several variables, from the textbook: exercises 3, 4, 5, p Solution of selected exercises on limits of functions of several variables, from the textbook: exercises 4, 5, 13 p Solution of exercises on single-variable differentiation: derivative of f(x) = exp(x); exercises, derivative of f(x) = tan(x), derivative of f(x) = e x2 /2. (16/10/2017) 5. Partial differentiation. Partial differentiation: definition and notation ( f xf, x, f x, f 1 ); extension of standard single-variable differentiation laws (sum, difference, etc.) to partial differentiation; 5

6 examples of partial derivatives from the textbook, paragraph 12.3, examples 1-3, p ; partial differentiation of the composition g(x, y) = f[u(x, y), v(x, y)] (chain rule); partial differentiation of the composition g(x, y) = f[u(x, y)] (chain rule); (other than in 1-d case...) existence of partial derivatives does not 2xy imply continuity (ex.gr. z defined as x 2 +y outside the origin and 0 2 in the origin; see exercise 36 from the textbook, p. 688); (as in 1-d case...) continuity does not imply the existence of the partial derivatives (ex.gr. z = x 2 +y 2 ). Higher-order partial derivatives: definition and notation; theorem on equality of mixed partials (statement only); examples of partial derivatives from the textbook, paragraph 12.3, examples 1-2, p Introduction to O(x) and o(x) notation. (17/10/2017) 6. Differentiability and linearization. O(x) and o(x) notation. Differentiability: definition of differentiability; differentiability in a point implies continuity in that point; existence of partial derivatives, continuity, differentiability; theorem: if the partial derivatives of a function are continuous in a neighbourhood of a point then the function is differentiable in that point, and thus also continuous (statement only; ex.gr. z = 2xy x 2 +y 2 outside the origin and z = 0 in the origin). Linearizations (linear approximations): 1-d case; 2-d case (and, more generally, n-d case); differentiability as the possibility of linearizing (linearly approximating); example from the textbook, paragraph 12.6, exercise 5, p

7 (17/10/2017) E3. Exercise class on partial differentiation and linearization, and on single-variable differentiation. Solution of selected exercises on partial differentiation from the textbook: exercise 7, p. 687; exercise 9, p. 687; exercise 22, p. 687; exercise 5, p Solution of selected exercises on linearization from the textbook: example 1, p. 704; exercise 5, p Solution of exercises on single-variable differentiation: inverse function theorem (explained by examples), case of ln(x), case of x, case of arctan(x) (left as a homework). (23/10/2017) 7. Gradient, differential, directional derivative. Gradient: definition; linearization expressed in terms of gradient. Differential: definition; linearization expressed in terms of differential. Directional derivative: definition of directional derivative (or rate of change along a given direction); directional derivative expressed in terms of gradient; geometrical interpretation of the gradient. Examples from the textbook: paragraph 12.6, exercise 1, p. 712; paragraph 12.7, example 2, p ; paragraph 12.7, example 3, p. 718; paragraph 12.7, example 4, p ; 7

8 paragraph 12.7, exercise 12, p (24/10/2017) 8. Extreme values. Introduction to extreme values: local maxima and minima (from 1-d to n-d); examples, z = x 2 +y 2, z = 1 x 2 y 2, z = x 2 +y 2, z = xy, z = x 2 y 2. Necessary and sufficient conditions for the existence of extreme values: definition of critical (stationary) points, singular points, boundary points; theorem on the necessary conditions (with proof); conditions are not sufficient (counterexamples: saddle points like z = xy, z = x 2 y 2 ); theorem on the sufficient conditions (without proof). Determination of the characteristics of a critical point via Hessian matrix (2-variable case only): a summary on the definiteness of a matrix; definition of Hessian matrix; determination of the characteristics of a critical point out of the definiteness of the Hessian matrix (without proof); in the case of indetermination... (brute force) characterization of critical points via analysis of neighbourhoods; example, z = 2x 3 6xy +3y 2 (from the textbook; paragraph 13.1, example 7, p. 748). (24/10/2017) E4. Exercise class on: gradient, directional derivative; extreme values. Solution of a selected exercise on gradient and directional derivative from the textbook: 8

9 exercise 17, p Solution of selected exercises on extreme values (mostly from the textbook): examples, z = x 4 +y 4, z = x 3 +y 3 ; example 1, p. 751; example 2, p ; exercise 4, p. 756; (30/10/2017) 09. Curves. Lagrange multipliers. Curves: expression by means of functions of the kind g(x, y,...) = 0; displacement on a curve; a remarkable application: parabolic reflector. The method of Lagrange multipliers: heuristic justification of the method; general statement (without proof) of the method (F = f λg; F = 0; g 0, no endpoint of the curve); remarkable examples from geometry, shape of the rectangle that can be inscribed in a circle and has the largest area, distance of a point from a straight line. (31/10/2017) 10. Method of least squares. A summary of singlevariable integration. Fitting data to a straight line (linear regression), via method of least squares: merit function given by the sum of the squares of the residuals; χ 2 merit function in case of uncertainties on y, and evaluation of the straight line; example from the textbook (example 1, p. 767); short discussion on related topics, linear fit, nonlinear fit and the Levenberg-Marquardt algorithm. 9

10 Linear fit with a constant value: the weighted mean; the sample mean. A summary of single-variable integration: graphical interpretation and definition; definite and indefinite integral (aka primitive aka antiderivative); the fundamental theorem of calculus; not to be forgotten: the +c! a b f(x)dx = b a f(x)dx, b a f(x)dx = c a f(x)dx+ b c f(x)dx, b a [αf(x)+βg(x)]dx = α b a f(x)dx+β b a g(x)dx; indefinite integral of x n, sin(x), cos(x), e x, 1/x. (31/10/2017) E5. Exercise class on Lagrange multipliers, and on single-variable integration. Solution of selected exercises on Lagrange multipliers: shape of the box that can be inscribed in a sphere and has the largest volume; from the textbook, exercise 2, p. 764; from the textbook, exercise 22, p Solution of exercises on single-variable integration: exercise, indefinite integral of f(x) = x x2 +2; integration by parts; indefinite integral of f(x) = xsin(x), f(x) = xcos(x). (06/11/2017) 11. Metric spaces and function spaces: an introduction. The issue of approximating: an example in R 3 (and special case of orthonormal vectors); 10

11 possibility of extending the method to functions; the importance of defining a metric in order to approximate a function by a linear combination of other functions. Metric spaces: vector space; metric (distance) in a space; norm and related metric; example of R n, with a mention of L2 (Euclidean) and L1 ( cityblock ) norm (and metric). A simple function space: set of the functions that are continuous and bounded on an interval; definition of a norm (and thus a metric) via an integral; least-squares approximation of a function f(x) in the interval [0, a] by means of a polynomial p+qx; example f(x) = x 2 with a = 1. (07/11/2017) 12. Least-squares approximation of a function by means of polynomials. Gram-Schmidt orthonormalization method. A summary of single-variable integration. A summary of single-variable integration (and differentiation): derivative with respect to a limit of integration of a definite integral; De L Hôpital s rule; limit case for a 0 of the least-squares approximation of a function f(x) in the interval [0, a] by means of a polynomial px+q (see lecture 11.). Least-squares approximation of a function f(x) in the interval [a, b] by meansofalinearcombinationofnorthonormalfunctions{φ 0 (x),..., φ n (x)}: generalization of last lecture s result: distance between a function f(x) and a linear combination p 0 φ 0 (x)+p 1 φ 1 (x); definition of a scalar product of two functions and its properties; expression of the distance between a function f(x) and a linear combination p 0 φ 0 (x)+p 1 φ 1 (x) by using the scalar product; the need of an orthonormal set of functions; from the case p 0 φ 0 (x)+p 1 φ 1 (x) to the general solution, 11

12 f approx (x) = n i=0 p iφ i (x), p i =< φ i (x) f(x) >. Gram-Schmidt method for the generation of an orthonormal base of functions: discussion of the method in R 3 ; generation of an orthonormal set of polynomials of degree 0, 1, 2 in [0, 1]. (07/11/2017) E6. Exercise class on approximation of a function by polynomials, and on single-variable integration. Approximation of functions by polynomials: normalization of the second-degree base polynomial in [0, 1], x 2 x+ 1 6 ; least-squares approximation of the function y = x 3 by a seconddegree polynomial in [0, 1]; least-squares approximation of the function e x by a second-degree polynomial in [0, 1]. Solution of exercises on single-variable integration: integration by substitution; normalization of the second-degree base polynomial in [0, 1], x 2 x+ 1 6, carried out by substituting x 1 2 with y; recursive calculation of the integrals 1 0 xn e x dx. (13/11/2017) 13. Complementary topics: Taylor and Maclaurin expansion. Fourier series (part 1 of 2). A mention of Taylor and Maclaurin expansion: general expression; examples of expx, cosx, sinx. Fourier orthonormal base in [ π, π]: scalar product (with normalization 1/π) within the interval [ π, π]; description of the orthonormal base; plot of the first functions. Expansion in Fourier series and Fourier theorem: 12

13 Fourier series; Fourier expansion of a function defined in [ π, π), and Fourier coefficients; example of unlimited total fluctuations, sin(1/x) in x = 0; Fourier theorem (without proof) for a function defined in [ π, π), with integrable modulus and limited total fluctuations; case of a continuous function. Further aspects of Fourier series: periodic extension; odd and even functions, decomposition of a function in an odd and an even part, Fourier coefficients. (14/11/2017) 14. Fourier series (part 2 of 2). Fourier series for a periodic function with generic period T: new scalar product and orthonormal base; angular frequency ω 2π/T. A remarkable example: Fourier expansion of a square wave (sign(x) = x /x). Continuity and rate of convergence of the Fourier coefficients: discontinuity and continuity of the periodic extension and its derivatives (up to the k-th), and n-dependency of the Fourier coefficients (O(1/n), O(1/n k+2 )); example of the square wave. Parseval s theorem for a Fourier series: proof of the theorem; example of the square wave (and a first evaluation of π, upon a discussion on the determination of e). Integration and differentiation of a Fourier expansion: integrability; differentiability and related condition on the continuity of the periodic extension; example of the square wave, non differentiability, integration between 0 and x and Fourier expansion of the func- 13

14 tion x. (14/11/2017) E7. Exercise class on Fourier series, the solution of the Basel problem, and on single-variable integration. Fourier expansion of y = x (base period [ π, π)): rate of convergence of the Fourier coefficients; evaluation of the Fourier coefficients; applicationofparseval stheoremandevaluationof n=1 1/n2 (Basel problem). Fourier expansion of y = x 2 (base period [ π, π)): rate of convergence of the Fourier coefficients; evaluation of the Fourier coefficients; application of Parseval s theorem and evaluation of n=1 1/n4 ; evaluation of the Fourier expansion of y = x via differentiation of the Fourier expansion of y = x 2 ; mention of the evaluation of the Fourier expansion of y = x 2 via integration of the Fourier expansion of y = x. Solution of exercises on single-variable integration: calculation of the integral e βx cosαxdx; calculation of the integral e βx sinαxdx. (20/11/2017) 15. Multiple integration (part 1 of 3). Introduction to double integration: heuristic interpretation as a volume; definition as a limit (relying on Riemann sums). Integration domains: simple and regular domains; integrability of a bounded, continuous function on a bounded, regular domain (without proof); improper integrals. Iterated integrals: iteration in the case of simple domains; 14

15 examples from the textbook (in all three cases both iterations of a simple domain were used), paragraph 14.2, example 1, p , paragraph 14.2, example 2, p , exercise 9, p (21/11/2017) 16. Fourier series (part 3 of 3). Complex numbers. Fourier analysis of the response to a periodic excitation of a low-pass filter system, described by the differential equation ẏ +y/τ = f(t)/τ: general solution; solution in the case of a constant excitation; solution in the case of a square wave excitation. Complex numbers: definition, any real number is a complex mumber, a complex mumber i exists i 2 = 1, anycomplexnumberzcanbewrittenasz = x+iy, withx, y R (x, y being the real and the imaginary part of z, respectively), the ordinary arithmetic properties of addition and multiplication are conserved; set C of the complex numbers; graphical representation (Argand-Gauss plane); complex conjugate, definition of z, z x iy, graphical representation, evaluation of the real and the imaginary part of a complex number, Rez = x = z+ z 2, Imz = y = z z 2i, complex conjugate of addition, subtraction, multiplication, z ± v = z ± v, z v = z v; modulus, definition z, z z z = x 2 +y 2, graphical representation, modulus of the complex conjugate, z = z, modulus of multiplication, z w = z w ; 15

16 division of two complex numbers, via solution of a system, v/z = v z/ z 2, example: calculation of (2 + 3i)/(4 2i); Euler s formula, derivation from Maclaurin expansions of e x, cos(x), sin(x), example: calculation of i (via system of two variables x and y, and via Euler s formula), representationofacomplexnumberz as z e iθ,withθ = arctan(rez/imz). (21/11/2017) E8. Exercise class on multiple integration via iterated integrals. Solution of selected exercises on double integration from the textbook: example 3, p. 800 (both iterations were, at least tentatively, used); exercise 15, p. 802 (both iterations were, at least tentatively, used); example 3, p. 804 (improper integral due to unbounded function); example 1, p. 803 (improper integral due to unbounded domain); exercise 2, p. 807 (improper integral due to unbounded domain). Volume of a right pyramid with a square base. (27/11/2017) 17. Multiple integration (part 2 of 3). Area and volume evaluation: area evaluation via D da 1; evaluation of the surface area of a circle; volume evaluation as an integral on a surface f(x, y) via da f(x, y); D evaluation of the volume of a sphere; extension of the concepts of double integration to the case of more than two variables; volume evaluation as a volume integral dv 1. D From Cartesian coordinates to polar, cylindrical and spherical ones: 16

17 polar coordinates, transformation and inverse transformation, area element; cylindrical coordinates, transformation and inverse transformation, volume element; spherical coordinates, transformation and inverse transformation, volume element, area element at fixed radius; evaluation of the surface area of a circle via da 1 and polar coordinates; D evaluation of the volume of a sphere via volume integral DdV 1 and spherical coordinates; evaluation of the surface area of a sphere via spherical coordinates. (28/11/2017) 18. Multiple integration (part 3 of 3). Evaluation of the Gaussian integral. Coordinate transformation (mapping) and Jacobian determinant: Jacobian determinant; Jacobian determinant of the inverse transformation; trasformation from Cartesian to polar coordinates and from polar coordinates to Cartesian, evaluation of (x,y) (ρ,θ), evaluation of (ρ,θ) (x,y), (x,y) = 1/ (ρ,θ). (ρ, θ) (x, y) Surface area of an elliptical disk. Example from the textbook (paragraph 14.4, example 8, p. 815). (28/11/2017) E9. Exercise class on change of variables in double integration. Solution of selected exercises on change of variables in double integration from the textbook: 17

18 exercise 5, p. 817; exercise 6, p. 817; exercise 4, p. 817; exercise 5, p. 817, with y 2 instead of x 2 ; exercise 33, p. 817; exercise 34, p (04/12/2017) 19. A classification of differential equations. Boundary conditions. Linear, scalar, ordinary differential equations. Definition of differential equations (DE). A classification of DE: ordinary and partial DE; scalar and vectorial DE; order of a DE; The further discussion is restricted to scalar, ordinary DEs (scalar ODEs) F(x, y, y,... y (n) ) = 0; linear and nonlinear DE; homogeneous and nonhomogeneous DE. Order of a DE and boundary conditions: number of unknown constants to be expected when solving a DE, i.e. number of boundary (initial) conditions required, equal to the order of the DE (without proof); Cauchy problem. Linear, scalar ODEs of any order: expression, n k=0 a k(x)y (i) (x) = f(x); homogeneous case, and linear combination of two solutions; nonhomogeneous case, and sum of a solution of this case with a solution of the homogeneous case; solution of linear, constant-coefficients, homogeneous, scalar 18

19 ODEs via auxiliary equation (without proof); examples from the textbook: paragraph 17.5, examples 1a, 1b, p (05/12/2017) 20. First-order scalar, ordinary differential equations. First-order, separable, scalar ODEs: description (DE can be nonlinear) and solution; examples from the textbook, paragraph 7.9, animal population growth example, paragraph 7.9, example 1, p. 446, paragraph 7.9, example 2, p First-order linear, homogeneous, scalar ODEs: description and solution as a separable DE; solution in the case of constant coefficients, also by means of the auxiliary equation; a remarkable example, dy dt + y τ = 0, decay time and half-life. First-order linear, nonhomogeneous, scalar ODEs: nonhomogeneous case solved by using the solution of the homogeneous case; examples from the textbook, paragraph 7.9, example 7, p , paragraph 7.9, example 8, p (05/12/2017) E10. Exercise class on first-order scalar, ordinary differential equations. Solution of selected exercises on first-order scalar ODEs from the textbook: exercise 3, p. 452; exercise 2, p. 452; exercise 8, p. 452; exercise 10, p. 452; exercise 13, p. 452; exercise 16, p

20 Partial fraction decomposition. (11/12/2017) 21. Second-order linear, scalar, ordinary differential equations with constant coefficients. Second-order linear, constant-coefficients, homogeneous, scalar ODEs: solution via auxiliary equation; example from the textbook (paragraph 3.7, example 3, p. 205); oscillations; example from the textbook (paragraph 3.7, example 5, p. 207). Solution of second-order linear, constant-coefficients, nonhomogeneous, scalar ODEs via method of variation of constants and Wronskian determinant: method of variation of constants, Wronskian determinant; example from the textbook (paragraph 17.6, example 1, p. 962). Solution of second-order linear, constant-coefficients, nonhomogeneous, scalar ODEs via method of undetermined coefficients : discussion in the case of a polynomial nonhomogeneous term, example from the textbook (paragraph 17.6, example 1, p. 962; see above), example above with x 2 instead of 4x as nonhomogeneous term; summary of the method in the case of polynomial, exponential, and sinusoidal nonhomogeneous terms (also when these terms are proportional to a solution of the associated homogeneous equation) [*], example above with e x instead of 4x as nonhomogeneous term. [*] discussed in lecture 22. (12/12/2017) 22. Linearity and solution of differential equations. Numerical methods for the solution of ordinary differential equations. Exploitation of linearity in the solution of second-order linear, constant-coefficients, nonhomogeneous, scalar ODEs where the nonhomogeneous term is given by a linear combination of different terms: example from the textbook (paragraph 17.6, example 1, p. 962) with 4x 2 4x instead of 4x as nonhomogeneous term. 20

21 Euler method (first-order Runge-Kutta method) for the numerical integration of first-order scalar ODEs. Runge-Kutta method for the numerical integration of first-order scalar ODEs: discussion of second-order Runge-Kutta method; mention of fourth- and higher-order Runge-Kutta methods, as well as of the availability of numerical libraries (ex.gr. GSL Gnu Scientific Library). Generalization of Runge-Kutta methods from the scalar to the vectorial case. Reduction of an n-th order vectorial (or scalar) ODE to a first-order vectorial one: discussion by using the numerical solution of the two-body gravitational interaction (ex.gr. orbit of the earth around the sun) as an example. (12/12/2017) E11. Exercise class on second-order linear, scalar, ordinary differential equations with constant coefficients. Solution of selected exercises on second-order linear, constant-coefficients, scalar ODEs from the textbook: examples 2a, 2b, 2c, p , exercises 11, 12, p. 967, solved via method of variation of constants and Wronskian determinant (examples 2a, 2b, each single term of exercise 11), method of undetermined coefficients (examples 2a, 2b, each single term of exercise 11, exercise 12), exploitation of linearity (example 2c, exercise 11). (18/12/2017) 23. Summary of the course (part 1 of 2). Summary of differential calculus of functions of two or more variables: differentiability and related theorem(statement only: if partial derivatives of a function exist in a neighbourhood of a point and are continuous in that point, then the function is differentiable in that point); gradient, differential, directional derivative; linearization / linear approximation; extreme values (critical, singular, boundary points), Hessian matrix 21

22 H = 2 z x 2 2 z y x. 2 z x y 2 z y 2 ;.... Lagrange multipliers; examples from past exams: exercise 1 of exam, exercise 2 of Mathematical Methods for Physics exam. Summary of integral calculus of functions of two or more variables: multiple integration via iterated integrals; change of variables in double integration, Jacobian determinant, Jacobian determinant of the inverse transformation x x u v (x,y,...) (u,v,...) = det y y u v ;..... examples from past exams: exercise 4 of exam, exercise 5 of exam (with an additional variation, i.e. the integration of x 2 +y on the unitary circle. (19/12/2017) 24. Summary of the course (part 2 of 2). Summary of integration of scalar ODEs: Cauchy problems; linear, constant-coefficients, homogeneous, scalar ODEs (solution via auxiliary equation); first-order scalar ODEs, separable DE, linear, homogeneous DE, linear, nonhomogeneous DE; second-order linear, constant-coefficients, scalar ODEs, homogeneous DE, solution of nonhomogeneous DE via method of variation of constants and Wronskian determinant, solution of nonhomogeneous DE via method of undetermined coefficients ; exploitation of linearity; 22

23 examples, mainly from past exams: y 3y +3y y = 0, exercise 2 of exam, from the textbook, paragraph 7.9, example 8, p. 450, y 2y +y = te t, exercise 3 of exam (with an additional variation, i.e. the integration of the DE with the nonhomogeneous term given by αe 2x +βsinx). (19/12/2017) E12. Summary exercises. Solution of selected exercises from past exams: (morning session) exercise 1e of Mathematical Methods for Physics exam, exercises 1-5 of exam; (afternoon session) exercise 1e of Mathematical Methods for Physics exam, exercises 1-5 of exam. (18/01/2018) Supplementary exercise class. Duration: 2h30 Supplementary exercise class: solution of selected exercises: exercises 1-5 of exam. (25/01/2018) Supplementary exercise class. Duration: 2h30 Supplementary exercise class: solution of selected exercises: exercises 1-5 of exam. 23