x^3 y^'''+x y^'-y=3-lnx, x>0 ; y(1)=3, y^'(1)=3, y^''(1)=0 ; yp=lnx ; {x, x lnx, x(lnx)^

X^3 y^'''+x y^'-y=3-lnx, x>0 ; y(1)=3, y^'(1)=3, ...

Key Concepts

Linear Differential Equations

This concept deals with equations where the dependent variable and its derivatives appear linearly. In the given problem, the differential equation features a combination of y and its derivatives multiplied by functions of x, establishing a framework in which standard linear solution techniques apply.

Non?Homogeneous Differential Equations

A non?homogeneous differential equation includes a forcing term—an expression that is not identically zero—which makes the solution different from the complementary (homogeneous) solution. The term on the right-hand side of the equation, 3 - ln x, denotes such an external influence, requiring the addition of a particular solution to the general homogeneous solution.

Particular and Complementary Solutions

The overall general solution of a non?homogeneous differential equation is composed of the complementary (or homogeneous) solution, which satisfies the associated homogeneous equation, plus a particular solution that specifically addresses the non?homogeneous part. This separation is crucial for constructing the full solution and for applying initial conditions effectively.

Method of Undetermined Coefficients

This method involves guessing a form for the particular solution based on the non?homogeneous term. In the problem, a guess for the particular solution is provided (yp = ln x), a strategy often used when the forcing term is of a form (such as a polynomial, exponential, logarithm, or trigonometric function) whose derivatives yield similar types of terms.

Repeated Roots and Reduction of Order

When solving the homogeneous part of a differential equation, repeated roots in the characteristic equation lead to a family of solutions that include logarithmic factors. The set of functions {x, x ln x, x(ln x)²} exemplifies this phenomenon, showing how additional factors like ln x arise to provide a fundamental set of linearly independent solutions when a root occurs with multiplicity greater than one.

Initial Value Problems

Initial value problems (IVPs) provide conditions that the solution must satisfy at a given point. In the context of differential equations, these conditions are used to determine the arbitrary constants in the general solution, ensuring its uniqueness and its relevance to the physical or geometric situation being modeled.

Transcript

00:01 All right, so this one, the equation we're given is y triple prime plus y prime over x squared minus y over x cubed is equal to 3 over x cubed minus the natural log of x over x cubed.

00:21 Okay, well, so to get the part a is just to get our general solution.

00:26 And to do that, we just multiply constants by our functions.

00:31 Set, right? and so the function set we are given then is going to say that our general solution would be of the form.

00:38 So we have our y of p here, our natural log of x, and then plus a constant times everything in the function set that we're given.

00:48 So we would have c1 times x plus c2 times x natural log of x plus c3 times x times the natural log of x squared.

01:11 Okay, well, so we have this, so let's take the derivative, and when we do that, we get 1 over x plus c1 plus c2, which is going to be the natural log of x plus 1, because we're taking the derivative of x times the natural log of x plus the natural log of x times the derivative of x.

01:49 Okay, and then we do c3 times this derivative here, and that's going to be the natural log of x squared, plus two times the natural lot of x.

02:13 All righty, so we're going to take the derivative again.

02:34 So double prime, it's going to give us negative x to the negative 2 power, plus c2 times 1 over x, plus c3 times 2 over x, the natural log of x, plus 2 over x.

03:08 Okay, and then we have some initial conditions.

03:11 So we know that y of 1 is equal to 3...

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