Can You Solve This Third Order Homogeneous Differential Equation?

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In summary, the conversation discusses finding the general solution to a third order homogeneous differential equation given one known solution. The solution involves assuming (m-5) is a factor of the characteristic equation, resulting in a thrice repeated root and a solution of y=c_1e^{5x}+c_2xe^{5x}+c_3x^2e^{5x}. However, since the original equation was not given, the validity of this solution is uncertain.
  • #1
Saladsamurai
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Is this diff eq solvable? !

Homework Statement


Find the general solution to the third order homogeneous diff eq if one solution is known to be:

[tex]x^2e^{5x}[/tex]


I was thinking of using reduction of order but I don't have the original equation!
 
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  • #2
Okay. Since it was given that the 3rd order diff eq is homogeneous, it is okay to assume that there is no particular solution to this, right?

And I can also assume that (m-5) is a factor of the characteristic equation. I am also assuming that it is a thrice repeated root, so [itex]y=c_1e^{5x}+c_2xe^{5x}+c_3x^2e^{5x} [/itex]

I am not so confident in this, since it is all based on assumption. Any thoughts on the validity of this?
 
  • #3
It might help a bit if you told us what the differential equation was.
 
  • #4
d_leet said:
It might help a bit if you told us what the differential equation was.

Read the OP. It has not been given.
 
  • #5
Saladsamurai said:
Okay. Since it was given that the 3rd order diff eq is homogeneous, it is okay to assume that there is no particular solution to this, right?

And I can also assume that (m-5) is a factor of the characteristic equation. I am also assuming that it is a thrice repeated root, so [itex]y=c_1e^{5x}+c_2xe^{5x}+c_3x^2e^{5x} [/itex]

I am not so confident in this, since it is all based on assumption. Any thoughts on the validity of this?

Assuming this is a linear homogeneous 3rd order diff eq with constant coefficients, then that is the case and the differential equation must be (D- 5)3y= y"'-15y"+ 75y'- 125y= 0. Of course, we were not told that.
 
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  • #6
So, as it stands, this problem has been worded incorrectly. That is what I thought. Thank you.
 

FAQ: Can You Solve This Third Order Homogeneous Differential Equation?

Is there a general method for solving differential equations?

Yes, there are several methods for solving differential equations, such as separation of variables, integrating factors, and substitution. However, the specific method used depends on the type of differential equation and its complexity.

Can all differential equations be solved analytically?

No, not all differential equations have analytical solutions. Some may require numerical methods for approximation, while others may not have a solution at all.

What is the difference between an ordinary and a partial differential equation?

An ordinary differential equation involves a single independent variable, while a partial differential equation involves multiple independent variables. The solution of a partial differential equation is a function of all the independent variables, while the solution of an ordinary differential equation is a function of a single independent variable.

How do initial and boundary conditions affect the solvability of a differential equation?

Initial and boundary conditions are necessary for solving a differential equation as they provide specific values for the unknown function and its derivatives. Without these conditions, there may be an infinite number of solutions or no solution at all.

Are there any software or tools available for solving differential equations?

Yes, there are various software and tools available for solving differential equations, such as Matlab, Mathematica, and Wolfram Alpha. These tools use numerical methods to approximate the solution to a given differential equation.

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