Topics Sections 3.1 to 3.6 The textbook topics, definitions, examples and theorems
Edwards-Penney Ch 3, 3.1 to 3.4 (16.5 K, txt, 04 Jan 2015)
Edwards-Penney Ch 3, 3.5 to 3.7 (17.7 K, txt, 02 Jan 2015)
Slides: Shock-less auto. Rolling wheel on a spring. Swinging rod. Mechanical watch. Bike trailer. Physical pendulum. Classification: critically, over and under damped Phase-amplitude conversions Trig right triangle formulas Triu identities: sin(a+b), others by tricks, even-odd Cafe door. Pet door. Beats. Undetermined coefficients. Resonance. Chapter 3 references. Sections 3.4, 3.5, 3.6. Forced/Unforced oscillations.
Slides: Resonance and undetermined coefficients, cafe door, pet door, phase-amplitude (178.0 K, pdf, 08 Mar 2014)
Slides: Electrical circuits (112.8 K, pdf, 19 Feb 2016)
Slides: Unforced vibrations 2008 (647.6 K, pdf, 27 Feb 2014)
Slides: Forced damped vibrations (263.9 K, pdf, 10 Feb 2016)
Slides: Forced vibrations and resonance, Millenium Bridge, Wine Glass, Tacoma Narrows (253.0 K, pdf, 08 Mar 2014)
Slides: Forced undamped vibrations (214.2 K, pdf, 03 Mar 2012)
Slides: phase-amplitude, cafe door, pet door, damping classification (136.0 K, pdf, 08 Mar 2014)
REVIEW: Undetermined Coefficients Which equations can be solved THEOREM. Solution y_h(x) is a linear combination of atoms. THEOREM. Solution y_p(x) is a linear combination of atoms. THEOREM. (superposition) y = y_h + y_p
Slides: Basic undetermined coefficients (2018) (147.6 K, pdf, 14 Feb 2018)
Slides: Variation of parameters (164.5 K, pdf, 03 Mar 2012) EXAMPLE. How to find a shortest expression for y_p(x) using Details for x''(t)+x(t) = 1+t the trial solution x(t)=A+Bt the answer x_p(t)=1+t. BASIC METHOD. Given a trial solution with undetermined coefficients, find a system of equations for d1, d2, ... and solve it. Report y_p as the trial solution with substituted answers d1, d2, d3, ... THEORY. y = y_h + y_p, and each is a linear combination of atoms. How to find the homogeneous solution y_h(x) from the characteristic equation. How to determine the form of the shortest trial solution for y_p(x) METHOD. A rule for finding y_p(x) from f(x) and the DE. Finding a trial solution with fewest symbols. Rule I. Assume the right side f(x) of the differential equation is a linear combination of atoms. Make a list of all distinct atoms that appear in the derivatives f(x), f'(x), f''(x), ... . Multiply these k atoms by undetermined coefficients d_1, ... , d_k, then add to define a trial solution y. This rule FAILS if one or more of the k atoms is a solution of the homogeneous differential equation. Rule II. If Rule I FAILS, then break the k atoms into groups with the same base atom. Cycle through the groups, replacing atoms as follows. If the first atom in the group is a solution of the homogeneous differential equation, then multiply all atoms in the group by factor x. Repeat until the first atom is not a solution of the homogeneous differential equation. Multiply the constructed k atoms by symbols d_1, ... , d_k and add to define trial solution y. Explanation: The relation between the Rule I + II trial solution and the book's table that uses the mystery factor x^s. EXAMPLES. y'' = x y'' + y = x exp(x) y'' - y = x exp(x) y'' + y = cos(x) y''' + y'' = 3x + 4 exp(-x) THEOREM. Suppose a list of k atoms is generated from the atoms in f(x), using Rule I. Then the shortest trial solution has exactly k atoms. EXAMPLE. How to find a shortest trial solution using Rules I and II. Details for x''(t)+x(t) = t^2 + cos(t), obtaining the shortest trial solution x(t)=d1+d2 t+d3 t^2+d4 t cos(t) + d5 t sin(t). How to use dsolve() in maple to check the answer. EXAMPLE. Suppose the DE has order n=4 and the homogeneous equation has solution atoms cos(t), t cos(t), sin(t), t sin(t). Assume f(t) = t^2 + cos(t). What is the shortest trial solution? EXAMPLE. Suppose the DE has order n=2 and the homogeneous equation has solution atoms cos(t), sin(t). Assume f(t) = t^2 + t cos(t). What is the shortest trial solution? EXAMPLE. Suppose the DE has order n=4 and the homogeneous equation has solution atoms 1, t, cos(t), sin(t). Assume f(t) = t^2 + t cos(t). What is the shortest trial solution?