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Course

Subject:	Mathematics
Topic:	Differential Equations
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Name:	Math Doctor Bob (Robert Donley)
Type:	Individual

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Ordinary Differential Equations: Solutions and Examples

Video Lectures

Displaying all 41 video lectures.

I. First-Order ODEs
Lecture 1 Play Video	General Solution of y' + xy = 0 Calculus/ODEs: Find the general solution of the differential equation y' + xy = 0. Then find the specific solution where y(0) = -1. Use the specific solution to sketch all solutions in the xy-plane.
Lecture 2 Play Video	Verifying the Solution of an ODE Calculus/ODEs: Verify that y(x) = 2e^{2x} + 3e^{3x} is a solution to the ODE y"- 5y' + 6y = 0. Then sketch the graph.
Lecture 3 Play Video	The Logistic Function 1: Solving The ODE Calculus/ODEs: We introduce the logistic differential equation as a variation on exponential growth. We solve the differential equation using separation of variables and note general properties of the graph. A precise sketch of the S-curve occurs in Part 2.
Lecture 4 Play Video	The Logistic Function 2: Sketching The S-Curve Calculus/ODEs: In this part, we use the logistic differential equation to sketch the S-curve for the logistic growth function. The key step is identify the coordinates of the inflection point.
Lecture 5 Play Video	General Solution to y' - 3y = b(x) Calculus/ODEs: Find the general solution to y' - 3y = b(x) when (a) b(x) = e^x, (b) b(x) = x, and (c) b(x) = cos(x).
Lecture 6 Play Video	Example of Population Growth 1 Calculus/ODEs: A fish population doubles every 5 years. If the initial population is 200 fish, how many fish will there be in 6 years? How long will it take to attain a population of 32,000 fish?
Lecture 7 Play Video	Example of Population Growth 2 Calculus/ODEs: A fish population experiences exponential growth with an initial population of 5000 fish. If the population grows to 32,000 fish in 20 years, how long did it take for the population to double? We show an alternate solution using the equation A(t) = A0 2^{t/DP}.
Lecture 8 Play Video	Example of Radioactive Decay 1 Calculus/ODEs: Carbon-14 has a half-life of 5,715 years. A fossil has lost 75% of its original amount of C-14. How old is the fossil? How much C-14 remains after 50,000 years?
Lecture 9 Play Video	Example of Radioactive Decay 2 Calculus/ODEs: A sample contains 50 grams of radium-226. After 500 years, 6.5 grams remain. What is the half-life of Ra-226? We also give an alternate solution using the equation A(t) = A0 (1/2)^{t/HL}.
II. Second-Order ODEs
Lecture 10 Play Video	General Solution to y" - 6y' + 9y = 0 ODEs: Find all solutions to the ODE y" - 6y' + 9y = 0. The characteristic equation for this ODE has a double root. We use variation of constants to obtain the solution of the form y2 = xe^{3x}.
Lecture 11 Play Video	Complex Numbers for ODEs (1 of 4) ODEs: We define and present basic properties of the complex numbers. This part includes addition, subtraction, scalar multiplication, complex conjugate and modulus.
Lecture 12 Play Video	Complex Numbers for ODEs (2 of 4) ODEs: We describe basic properties of the complex numbers. A method for dividing complex numbers is given. Real and imaginary parts of a complex number are defined. Euler's formula is stated, and various consequences are interpreted.
Lecture 13 Play Video	Complex Numbers for ODEs (3 of 4) ODEs: We show the connection of the Euler formula to polar coordinates on the xy-plane. The geometric interpretation of multiplication is given, and we show how cosine and sine arise as solutions to y" + y = 0. Finally we give formulae for sine and cosine in terms of exp(itheta) and exp(-itheta) and interpret geometrically.
Lecture 14 Play Video	Complex Numbers for ODEs (4 of 4) ODEs: We connect cosine and sine to hyperbolic cosine and sine using the formula with exponentials. Basic properties of these hyperbolic trig functions are given, including their ODE: y" - y = 0.
Lecture 15 Play Video	General Solution to y" - 6y' +25y = 0 ODEs: Find the general solution to the linear ODE y" - 6y' + 25y = 0. The characteristic polynomial of this equation has roots r = 3 +/- 4i and solutions based on e^{3x}cos(4x) and e^{3x}sin(4x).
Lecture 16 $Antiderivative of e^{3x} cos(4x) (ODE Solution)$ Play Video	Antiderivative of e^{3x} cos(4x) (ODE Solution) ODEs: We find the antiderivative of e^{3x} cos(4x) by using the ODE y" - 6y' +25y = 0. It is more commonly solved by 2 integration by parts.
Lecture 17 Play Video	Antiderivative of x^2 e^x (ODE Solution) ODEs: We compute the antiderivative of x^2 e^x using the ODE y''' - 3y" + 3y' - y = 0. We normally solve this with two integration by parts.
Lecture 18 Play Video	General Solution of y'''-4y''+5y'-2y=0 ODEs: Find all solutions of the linear ordinary differential equation y'''-4y''+5y'-2y=0. Techniques include the rational root test and synthetic division. This equation has roots 2 and 1 (twice).
Lecture 19 $Wronskian for {e^{3x}, e^{-x}, 2}$ Play Video	Wronskian for {e^{3x}, e^{-x}, 2} ODEs: Show that the set of functions {e^{3x}, e^{-x}, 2} is a linearly independent set. These functions are in the solution space of y''' -2y'' - 3y' = 0. We show linear independence by computing the Wronskian of the set. We also show linear independence by solving a system of linear equations.
Lecture 20 $Linear Dependence of {x^2-1, x^2+x, x+1} Using Wronskian$ Play Video	Linear Dependence of {x^2-1, x^2+x, x+1} Using Wronskian ODEs: Consider the set of functions S = {x^2-1, x^2+x, x+1}. Is S a linearly dependent set? If not, find a relation in S. We test linear independence by computing the Wronskian.
Lecture 21 Play Video	Annihilator Method 1: Real Linear Factors ODEs: We use the annihilator method to solve y" - 2y' - 3y = b(x), where b(x) = (a) e^{2x}, (b) e^{3x}, and (c) e^{3x} + e^{-x}. We explain the method as an application of characteristic polynomials.
Lecture 22 Play Video	Example of Annihilator Method: y"-y = sin(2x) ODEs: Using the annihilator method, find all solutions to the linear ODE y"-y = sin(2x).
Lecture 23 Play Video	Power Series Solution for y"-2y'+y=x, y(0)=0, y'(0)=1 ODEs: Find the first four terms of the power series solution to the IVP y"-2y'+y=x, y(0)=0, y'(0)=1. To check our answer, we find the solution using the annihilator method and expand the exponential terms into power series.
III. Applications of ODEs: Mass-Spring Systems
Lecture 24 Play Video	Mass-Spring Systems 1: Undamped Motion ODEs: We consider the mass-spring system governed by the IVP x" + 4x = 0, x(0) = 1, x'(0)=0. After finding the solution, we plot the trajectory in the phase plane. Then we generalize the method to find all trajectories in the phase plane.
Lecture 25 Play Video	Mass-Spring Systems 2: Underdamped Motion ODEs: Consider the mass spring system governed by the IVP x"+2x'+5x = 0, x(0)=1, x'(0)=0. Using the solution to the IVP, we describe the motion of the system and plot the trajectory in the phase plane. Then we use certain solutions to describe the entire phase portrait. We also note the connection to the slope field.
Lecture 26 Play Video	Mass-Spring Systems 3: Critically Damped Motion ODEs: Consider spring motion governed by the IVP x"+4x' +4x = 0, x(0) = 1, x'(0) = 0. From the solution, we describe the motion of the spring and plot the trajectory in the phase plane. In this case, there is no oscillation and exactly one dimension of eigenvectors.
IV. The Laplace Transform
Lecture 27 Play Video	Laplace Transform of f(t) = 2t-1 ODEs: Compute the Laplace transform of f(t) = 2t -1. We verify the answer using the IVP y'=2, y(0) = -1. Then we derive the general formula for the Laplace transform of t^n.
Lecture 28 Play Video	Laplace Transform of f(t) = sin(2t) ODEs: Compute the Laplace transform of f(t) = sin(2t). This requires finding an antiderivative of e^ax sin(bx); we note a quick solution based on a partial formula. We verify our answer using the IVP y"+4y=0, y'(0) = 2, y(0)=0. Finally we note the general formulas for sin(at) and cos(at).
Lecture 29 Play Video	Laplace Transform of f(t) = t sin(2t) ODEs: Find the Laplace Transform of f(t) = t sin(2t). We use the derivative formula for the Laplace Transform: L(y') = sL(y) - y(0) instead of the definition. This method also finds the Laplace transform of g(t) = t cos(2t). We verify the answer by finding the IVP with f(t) as the unique solution.
Lecture 30 $Laplace Transform of f(t) = e^{3t}cos(4t)$ Play Video	Laplace Transform of f(t) = e^{3t}cos(4t) ODEs: Using the First Shift Formula, find the Laplace Transform of f(t) = e^{3t}cos(4t). We verify our solution by applying the LT to the IVP for f(t).
Lecture 31 $Laplace Transform of f(t) = t^2 e^{2t} cos(3t)$ Play Video	Laplace Transform of f(t) = t^2 e^{2t} cos(3t) ODEs (Requires Complex Numbers): Find the Laplace transform of the function f(t) = t^2 e^{2t} cos(3t). We give a heuristic method that combines the First Shift Formula and Euler's formula with the rule for L(t^n). This method applies in general to f(t) = t^n e^{at} cos(bt) and g(t) = t^n e^{at} sin(bt).
Lecture 32 Play Video	Inverse Laplace Transform of (s-1)/s^2(s^2+4) ODEs: Find the inverse Laplace transform of L[f(t)](s) = (s-1)/s^2(s^2+4). We use partial fractions to expand into terms where the inverse Laplace transform is recognizable.
Lecture 33 $Laplace Transform Solution of y'-3y=e^{2t}, y(0)=2$ Play Video	Laplace Transform Solution of y'-3y=e^{2t}, y(0)=2 ODEs: Using the Laplace Transform, find the solution to the IVP y'-3y=e^{2t}, y(0)=2. We note how one uses the Inverse Laplace Transform to attain a solution.
Lecture 34 Play Video	Laplace Transform Solution of y"-2y'-3y=e^t, y(0) = 0, y'(0) = 1 ODEs: Using the Laplace Transform, find the solution of the IVP y"-2y'-3y=e^t with y(0) = 0, y'(0) = 1. We compute the Laplace transform of f(t) = e^at and derive the formula L(y') = sL(y) - y(0), which plays a central role in the solution.
Lecture 35 Play Video	Laplace Transform of f(t) = 2 on the Interval (1,2) ODEs: Find the Laplace transform of the step function f(t) = 2 on the interval (1,2) and zero elsewhere. We rewrite f(t) as a difference of shifted unit step functions, and then apply the Second Shift Formula.
Lecture 36 Play Video	Second Shift Formula for a Piecewise-defined Function ODEs: Find the Laplace transform of the piecewise-defined function f(t) = t on (0,1), 1 on (1,3) , and 4-t on (3,4). We proceed by rewriting f(t) in terms of shifted unit step functions and then apply the Second Shift Formula. We check our work using integration by parts.
Lecture 37 Play Video	Laplace Transform Solution of y'-y=f(t) (Piecewise-Defined) ODEs: Use the Second Shift Formula for the Laplace Transform to solve the IVP y'-y=f(t), where f(t) = t on the interval [1,2), = 0 elsewhere. The key step here is to invert the Second Shift Formula to obtain y(t).
V. The Convolution Theorem & Fourier Series
Lecture 38 Play Video	Example of Convolution Theorem: f(t)=t, g(t)=sin(t) ODEs: Verify the Convolution Theorem for the Laplace transform when f(t) = t and g(t) = sin(t). The Convolution Theorem states that L(f*g) = L(f) . L(g); that is, the Laplace transform of a convolution is the product of the Laplace transforms.
Lecture 39 Play Video	Convolution Theorem for y'-2y=e^t, y(0)=0 ODEs: Using the Convolution Theorem, find the solution to the IVP y'-2y=e^t, y(0)=0. We solve by using the Laplace Transform; the Convolution Theorem is used instead of partial fractions.
Lecture 40 Play Video	Fourier Series: Example of Orthonormal Set of Functions Differential Equations: Prelude to Fourier series. Show that the sets B1 = {1, sqrt(2) cos(x), sqrt(2) sin(x)} and B2 = {1, exp(-ix), exp(ix)} are orthonormal sets of functions with respect to the inner product (f, g) = 1/2pi int f(x) {bar g(x)} dx. Then verify Parseval's Identity for f(x) = sin(x) with respect to each set.
Lecture 41 Play Video	Fourier Series: Example of Parseval's Identity Differential Equations: Find the Fourier coefficients of the square wave function f(x) = -1 on the interval (-pi, 0), 1 on the interval (0, pi). Then state Parseval's Identity in this case. With this, we show that sum 1/n^2 = pi^2/6.