- Chapter 1: First-order differential equations
- Chapter 1.10: The existence-uniqueness theorem; Picard iteration
- Chapter 1.11: Finding roots of equations by iteration
- Chapter 1.12: Difference equations, and how to compute the interest due on your student loans
- Chapter 1.13: Numerical approximations; Euler's method
- Chapter 1.14: The three term Taylor series method
- Chapter 1.15: An improved Euler method
- Chapter 1.16: The Runge-Kutta method
- Chapter 1.17: What to do in practice
- Chapter 1.2: First-order linear differential equations
- Chapter 1.3: The Van Meegeren art forgeries
- Chapter 1.4: Separable equations
- Chapter 1.5: Population models
- Chapter 1.6: The spread of technological innovations
- Chapter 1.7: An atomic waste disposal problem
- Chapter 1.8: The dynamics of tumor growth, mixing problems and orthogonal trajectories
- Chapter 1.9: Exact equations, and why we cannot solve very many differential equations
- Chapter 2: Second-order linear differential equations
- Chapter 2.1: Algebraic properties of solutions
- Chapter 2.1 1: Differential equations with discontinuous right-hand sides
- Chapter 2.10: Some useful properties of Laplace transforms
- Chapter 2.12: The Dirac delta function
- Chapter 2.13: The convolution integral
- Chapter 2.14: The method of elimination for systems
- Chapter 2.15: Higher-order equations
- Chapter 2.2: Linear equations with constant coefficients
- Chapter 2.3: The nonhomogeneous equation
- Chapter 2.4: The method of variation of parameters
- Chapter 2.5: The method of judicious guessing
- Chapter 2.6: Mechanical vibrations
- Chapter 2.7: A model for the detection of diabetes
- Chapter 2.8: Series solutions
- Chapter 2.9: The method of Laplace transforms
- Chapter 3.1: Algebraic properties of solutions of linear systems
- Chapter 3.1 1: Fundamental matrix solutions; e *'
- Chapter 3.10: Equal roots
- Chapter 3.12: The nonhomogeneous equation; variation of parameters
- Chapter 3.13: Solving systems by Laplace transforms
- Chapter 3.2: Vector spaces
- Chapter 3.3: Dimension of a vector space
- Chapter 3.4: Applications of linear algebra to differential equations
- Chapter 3.5: The theory of determinants
- Chapter 3.6: Solutions of simultaneous linear equations
- Chapter 3.7: Linear transformations
- Chapter 3.8: The eigenvalue-eigenvector method of finding solutions
- Chapter 3.9: Complex roots
- Chapter 4.1: Introduction
- Chapter 4.1 1: The principle of competitive exclusion in population biology
- Chapter 4.12: The Threshold Theorem of epidemiology
- Chapter 4.13: A model for the spread of gonorrhea
- Chapter 4.2: Stability of linear systems
- Chapter 4.3: Stability of equilibrium solutions
- Chapter 4.4: The phase-plane
- Chapter 4.5: Mathematical theories of war
- Chapter 4.6: Qualitative properties of orbits
- Chapter 4.7: Phase portraits of linear systems
- Chapter 4.8: Long time behavior of solutions; the PoincarLBendixson Theorem
- Chapter 4.9: Introduction to bifurcation theory
- Chapter 5.1: Two point boundary-value problems
- Chapter 5.3: Introduction to partial differential equations
- Chapter 5.4: Fourier series
- Chapter 5.5: Even and odd functions
- Chapter 5.6: Return to the heat equation
- Chapter 5.7: The wave equation
- Chapter 5.8: Laplace's equation
Differential Equations and Their Applications: An Introduction to Applied Mathematics 3rd Edition - Solutions by Chapter
Full solutions for Differential Equations and Their Applications: An Introduction to Applied Mathematics | 3rd Edition
Differential Equations and Their Applications: An Introduction to Applied Mathematics | 3rd Edition - Solutions by ChapterGet Full Solutions
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.
A symmetric matrix with eigenvalues of both signs (+ and - ).
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.
Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.
Every v in V is orthogonal to every w in W.
Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·
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