General Solution of inhomogeneous ODE

In summary, the conversation discusses solving a second order non-homogeneous differential equation using three particular solutions (Y1 = 1, Y2 = x, Y3 = x^2) to find a general solution. The approach is to find the difference of two of the solutions and use them to create three separate homogeneous equations. However, the person is having trouble solving for the coefficients a(x), b(x), and c(x) in the homogeneous equation. They are seeking advice or alternative methods to solve the system effectively.
  • #1
Just_some_guy
16
0
I am having a little trouble with a problem I am trying to solve.

Given three particular solutions

Y1(x)= 1, Y2(x)= x and Y3(x)= x^2

Write down a general solution to the second order non homogeneous differential equation.

What I have done so far is to realize if Y1,2 and 3 are solutions then the difference of two of the solutions is a solution to the homogeneous equation. So I used Y3-Y1, Y3-Y2 and Y2-Y1 to give three separate homogeneous equations. At this point I am trying so solve for the coefficients a(x), b(x) and c(x) from

a(x)y'' + b(x)y' + c(x) y= 0

where y is the difference of two of the particular solutions.

However I can seem to solve this system effectively by equating like powers etc... I was hoping someone could offer some advise or even an alternative method. I have solved problems like this with only two solutions but the case was simple.Regards
Guy
 
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  • #2
In future posts, please do not delete the homework template. Its use is required for homework questions.
 
  • #3
Mark44 said:
In future posts, please do not delete the homework template. Its use is required for homework questions.

Apologies I didnt realize it was necessary
 

FAQ: General Solution of inhomogeneous ODE

1. What is the difference between a homogeneous and inhomogeneous ODE?

A homogeneous ODE is one in which the dependent variable and its derivatives appear only in linear combinations, while an inhomogeneous ODE has additional terms that are not dependent on the dependent variable or its derivatives.

2. How do you find the general solution of an inhomogeneous ODE?

The general solution of an inhomogeneous ODE can be found by adding the general solution of the corresponding homogeneous equation to a particular solution of the inhomogeneous equation. The particular solution can be found using methods such as variation of parameters or undetermined coefficients.

3. Can an inhomogeneous ODE have more than one particular solution?

Yes, an inhomogeneous ODE can have multiple particular solutions. This is because the particular solution is not unique and can be chosen based on the method used to find it.

4. How does the presence of initial conditions affect the general solution of an inhomogeneous ODE?

The presence of initial conditions helps to determine the constants in the general solution of an inhomogeneous ODE. These initial conditions can be used to find the particular solution and then the constants in the general solution can be determined to satisfy the initial conditions.

5. Is there a way to check if a particular solution of an inhomogeneous ODE is correct?

Yes, one can substitute the particular solution back into the original inhomogeneous equation and see if it satisfies the equation. If it does, then the particular solution is correct.

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