Y^''-6 y^'-4 y=4 sin3 t-e^3 t t^2+1 / t

Name: Problem 11, Chapter 4: Fundamentals of Differential Equations
Uploaded: 2019-10-13T01:57:34-08:00
Duration: 4 min 3 s
Channel: Brian Ketelobeter
Description: y^''-6 y^'-4 y=4 sin3 t-e^3 t t^2+1 / t

Key Concepts

Second Order Linear Differential Equations

These are differential equations where the highest derivative is the second derivative and the equation is linear in the unknown function and its derivatives. The general solution is typically the sum of the homogeneous solution (obtained by setting the forcing function to zero) and any particular solution that satisfies the entire nonhomogeneous equation.

Homogeneous Solution via the Characteristic Equation

For second order linear differential equations with constant coefficients, one finds the homogeneous solution by setting up and solving the characteristic (auxiliary) equation. The roots of the resulting quadratic equation determine the form of the homogeneous solution, which can include real distinct, real repeated, or complex conjugate roots.

Particular Solution for Nonhomogeneous Equations

This involves finding a specific solution that satisfies the entire nonhomogeneous equation. The approach can vary depending on the form of the nonhomogeneous term. When the forcing function is composed of several different types of functions, separate particular solutions for each term may be found and then combined.

Method of Undetermined Coefficients

This technique is used to find particular solutions when the nonhomogeneous term is a linear combination of functions such as exponentials, polynomials, sines, and cosines. It involves assuming a form for the particular solution with unknown coefficients and then determining these coefficients by substituting the assumed solution into the differential equation.

Superposition Principle

Due to the linearity of the differential equation, the overall particular solution can be constructed as the sum of individual particular solutions, each corresponding to one of the terms on the right-hand side. This principle allows for the separate treatment of different types of forcing functions within the same differential equation.

Transcript

00:03 Okay, good day ladies and gentlemen.

00:07 Today we're looking at problem number 11 here and the question is whether or not we can apply the method of undetermined coefficients to this ordinary differential equation here.

00:23 And so i'm not really going to go through all the different parts of the undetermined coefficients because i think.

00:35 Think there is probably five to six distinct cases, but it is important, i think, for you to know each of those cases because they tell you how to go about solving the ordinary differential equations.

00:55 They'll tell you how to solve a bunch of cases or a bunch of ordinary differential equations, in particular ones with constant coefficients here.

01:09 And so the first thing is to realize that, oops, sorry about that.

01:16 So if i take three different distinct functions, i think f1 of t here, f2 of t here, and f3 of t, now i'm going to really look at this case by case in each case.

01:29 So in the first case, this one here is, and it is in fact proper form.

01:39 It is one of the cases, and i'm not sure which, but if you look, if you flip through, you'll see the thing cases.

01:47 And this is, in fact, one of the cases.

01:49 And the second one here, again, is also a case.

01:56 And again, i don't know exactly which one, but it is.

01:59 Is in fact one of the cases covered by the undetermined coefficients...

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