Determine the general solution to the given differential equation. Derive your trial solution us

Determine the general solution to the given differential...

Key Concepts

Linear Differential Equations

Linear differential equations are equations involving an unknown function and its derivatives where the function and its derivatives appear linearly (i.e., without being multiplied together or raised to a power). They can be homogeneous (equal to zero) or non-homogeneous (equal to some non-zero function), and their solutions are often expressed as the sum of a complementary solution and a particular solution.

Homogeneous and Non-homogeneous Equations

In the study of differential equations, a homogeneous equation is one in which the sum of the terms is set to zero, which allows for the determination of a complementary solution based solely on the differential operator. In contrast, non-homogeneous equations include an additional forcing term (or non-homogeneous term), and their general solution is formed by adding a particular solution that accounts for that forcing term to the complementary solution.

Characteristic Equation

The characteristic equation is derived from a linear differential equation with constant coefficients by assuming a solution of the form e^(rx). This substitution transforms the differential equation into an algebraic equation in terms of r. The roots of the characteristic equation indicate the form of the complementary solution of the homogeneous part of the differential equation.

Method of Annihilators

The annihilator method is a technique used to solve non-homogeneous linear differential equations by applying a differential operator that 'annihilates' the non-homogeneous term, transforming the original equation into a higher-order homogeneous equation. This approach helps in systematically finding a suitable trial solution for the particular integral.

Trial Solution

A trial solution is an assumed form of the particular solution that is substituted into the non-homogeneous differential equation to determine unknown coefficients. Its form is chosen based on the type of the forcing term and adjusted if the assumed form overlaps with the complementary solution to ensure linear independence.

General Solution

The general solution of a linear differential equation combines the complementary solution of the homogeneous equation and a particular solution of the non-homogeneous equation. This aggregate solution encompasses all possible solutions and satisfies the full differential equation.

Transcript

00:01 Okay, so for this problem, let's first rewrite this in differential operator form.

00:04 So that's going to be d cubed minus d squared plus d minus d or sorry, minus 1 times y is equal to 9e to the negative x.

00:16 So i'm going to attempt to factor this first here.

00:20 So to do so, i'm going to write out the coefficients like so and then try to, well, yeah, try to use the coefficients and factor.

00:31 Since the coefficient here is equal to 1, the rational roots are only going to be plus or minus 1.

00:37 So i'm going to try 1 here first.

00:39 So if i do that, then i multiply this 1 here.

00:43 So that becomes 0, 1 times 0 is 0, then add, bring down the 1.

00:49 Then 1 times 1 is also 1, and then 0.

00:52 Great.

00:53 So since this last number is 0, this is going to be a root of this equation here.

00:59 So we're going to have d minus 1.

01:02 Then this is going to be the coefficients of the remainder.

01:05 So then we have d squared plus 1, y, is equal to 9e to the negative x.

01:11 And then this doesn't factor anymore.

01:14 We also need to find the corresponding or the complementary solution.

01:21 So that's going to be when the right -hand side is equal to 0.

01:24 So y is c is equal to 0.

01:28 So now, to solve this, we find the corresponding auxiliary polynomial and set it equal to 0, so that's r minus 1, and then r squared plus 1, and that's equal to 0.

01:40 So from this part here, we get that r is equal to 1, and from this equation here, we'll get that r squared equals negative 1, or r is going to be equal to plus or minus i.

01:52 So our complementary solution, y of c of x, is going to take the 4 .5 .4.

01:59 C1e to the c1e to the x like so, then plus c2.

02:05 This is going to be now e to the 0, e to the a is 0, so that's 1.

02:11 And then we're going to have a sign x and then plus a c3 cosine of x.

02:18 Cosine of x.

02:20 Next we need to take care of this right hand side here.

02:24 So we have f of x is equal to 9e to the negative x.

02:29 So we need to find the annihilator of this function...