How Do You Determine p(x) and q(x) in a Second Order ODE with Given Solutions?

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SUMMARY

The discussion focuses on determining the coefficients p(x) and q(x) in a second-order ordinary differential equation (ODE) given the fundamental solutions x^2 + 2 and x^2 - 2. The correct approach involves substituting these solutions into the standard form of the ODE, y'' + p(x)y' + q(x)y = 0, and recognizing that the derivatives of both solutions are identical. The conclusion is that p(x) and q(x) cannot both be zero; instead, they must be solved as variables with respect to a constant x.

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brad sue
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Hi ,
please can I have your help?

x^2+2 and x^2-2 are fundamental set of solutions of a second order ODE. find the ODE.
form:y''+p(x) y'+ q(x) y=0.

I tried to replace the two solutions in the equation but because those solutions have exact same first and second derivatives, I found p(x)=q(x)=0 !

Can you help me to find p(x) and q(x)?
Thank you
B.
 
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Try again. Write out the two equations you get and solve for p and q as if they were variables and x was a constant. The two equations will be identical except for the factor in front of q (ie, y), since the two solutions only differ in their constant term (so that all their derivatives will be equal).
 

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