General Solution of inhomogeneous ODE

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SUMMARY

The discussion focuses on finding the general solution to a second-order non-homogeneous differential equation given three particular solutions: Y1(x) = 1, Y2(x) = x, and Y3(x) = x^2. The user identifies that the differences between these solutions yield homogeneous equations, leading to the formulation of a system involving coefficients a(x), b(x), and c(x) in the equation a(x)y'' + b(x)y' + c(x)y = 0. The user seeks advice on effectively solving this system, particularly in equating like powers of the solutions.

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Just_some_guy
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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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In future posts, please do not delete the homework template. Its use is required for homework questions.
 
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
 

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