Find a general solution [Diff Eq]

[jinksys]

Homework Statement

Find a general solution for y''' - 6y'' + 9y' = 0

Homework Equations

The Attempt at a Solution

I know that the general solution for a homogeneous DEQ is
Y(x) = c1y1(x) + c2y2(x) ... cnyn(x)

however, I am not given y1, y2 , or y3 so I am to assume that the answer is

y(x) = c1y1(x) +c2y2(x) + c3y3(x).

 

[Mark44]

Homework Statement

Find a general solution for y''' - 6y'' + 9y' = 0

Homework Equations

The Attempt at a Solution

I know that the general solution for a homogeneous DEQ is
Y(x) = c1y1(x) + c2y2(x) ... cnyn(x)

however, I am not given y1, y2 , or y3 so I am to assume that the answer is

y(x) = c1y1(x) +c2y2(x) + c3y3(x).

The whole point of solving a differential equation is to find the solutions. You are generally not going to be given them.

The first thing to do is to find the characteristic equation, and then solve it. Each solution r of your characteristic equation will lead to a function of the form e^rx.

 

[jinksys]

The whole point of solving a differential equation is to find the solutions. You are generally not going to be given them.

The first thing to do is to find the characteristic equation, and then solve it. Each solution r of your characteristic equation will lead to a function of the form e^rx.

This is a question from my first test in differential equations:

We hadn't covered second-order and higher DEQs at that time so this problem must be solvable without those techniques.

I'm just wondering how I would have solved this problem not knowing the characteristic equation, laplace transforms, etc.

 

[Mark44]

You don't need Laplace transforms, but that's a tough problem if you don't have any experience with second-order or third-order DEs. Your general solution will have the form y = c₁y₁(x) + c₂y₂(x) + c₃y₃(x), but it's hard for me to believe that this is all your instructor wanted.

The natural approach to this problem involves solving the characteristic equation, as well as knowing what needs to happen when there are repeated roots of the characteristic equation.

If I were you I would ask the professor for some guidance on this problem, given that you don't (or didn't) have the skills yet to solve the differential equation.