Can Elimination Method Reduce the Order of Differential Equations?

In summary, when solving differential homogeneous equations for non-trivial solutions, there are two methods that can be used: determinant and elimination. The determinant method results in an 8th order equation, while the elimination method results in a 10th order equation. However, the elimination method can also be used to eliminate variables and simplify the system, resulting in a lower-order equation. Taking the value of one variable from one equation and substituting it into the others is allowed in this process.
  • #1
adnan jahan
96
0
Dear Fellows, I want to solve 4 differential homogeneous equations for non trivial solution. I have found two methods
1) Determinent. 2) Elimination
By det. I got 8th order resultant equation and by second method I got 10th order. As each of initial equation is 2nd order so 8th is correct. But what goes wrong in Elimination I could not understand.
Example.
x+2Dy+D2z=0
9Dx+4y-6D3z=0
x+D2+Dz=0

determinant gives 5th order equation
while if we eliminate by taking value of z from eq i and put in ii and iii, and we will get iv , v. which finally gives 4th order equation.

I doubt in taking the value of z from first equation, is this thing allowed or not? if no then why?
 
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  • #2
Yes, it is allowed to take the value of z from the first equation and substitute it into the other equations. The reason why you get a lower-order equation when you eliminate variables is because you are simplifying the system by removing some variables. In your example, you are reducing a system of four second-order equations to a single fourth-order equation.
 

Related to Can Elimination Method Reduce the Order of Differential Equations?

What is the concept of "Solving ODEs by Elimination"?

"Solving ODEs by Elimination" is a method used to solve Ordinary Differential Equations (ODEs) by eliminating one of the variables in the equation. This involves using algebraic manipulation and substitution to reduce the equation to a simpler form that can be solved using basic integration techniques.

What types of ODEs can be solved using this method?

This method can be used to solve separable ODEs, which are equations where the dependent variable and independent variable can be separated on opposite sides of the equation.

What are the steps involved in solving ODEs by Elimination?

The steps involved in this method include identifying the type of ODE, separating the variables, integrating both sides of the equation, solving for the constant of integration, and finally, substituting the constant back into the solution to obtain the final answer.

What are the benefits of using this method to solve ODEs?

Using this method can make solving ODEs simpler and more efficient, as it reduces the equation to a form that can be solved using basic integration techniques. It also helps to avoid more complex methods such as using power series or numerical methods.

Are there any limitations to solving ODEs by Elimination?

Yes, this method can only be used for separable ODEs. Additionally, it may not work for all types of ODEs and may require additional steps or techniques to solve more complex equations.

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