Mathematica: 2nd order PDE variable coefficients

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SUMMARY

The discussion focuses on solving a second-order partial differential equation (PDE) with variable coefficients using Mathematica. The equation presented is a''[t] + B[t]*a'[t] - A[t]*a[t] == 0, with initial conditions a[0] = 10^-9 and a'[0] = 0. The user has successfully determined the variable coefficients A[t] and B[t] over the relevant time interval but encountered difficulties applying NDSolve for the solution of a[t]. The need for alternative methods or insights into using NDSolve effectively is emphasized.

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Mathematics students, researchers in applied mathematics, and professionals working with differential equations in Mathematica will benefit from this discussion.

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a\text{''}[t]+B[t]*a'[t]-A[t]*a[t]==0

a[0] = 10^-9
a'[0] = 0
a[t] = ?

The coefficients A and B are variable over time. I HAVE solved (experimental and theoretical values) for the values of A and B over the time interval of interest!

I attempted to solve for a[t] using NDSolve as one normally would when coefficients are not variable over time. No luck.

Any help would be greatly appreciated!
 
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