Analytically solving ODEs with non-constant coefficients for a specific t

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SUMMARY

The discussion centers on solving ordinary differential equations (ODEs) with non-constant coefficients, specifically in the form of f(t)y'' + g(t)y' + h(t)y = 0. It concludes that simply substituting a specific value of t into the ODE for an analytical solution is not reliable, as it may yield a coarse approximation that lacks accuracy. The conversation emphasizes the importance of understanding the time-varying nature of vector fields represented by differential equations and suggests that the method of integrating factors may be a viable approach for finding solutions.

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  • Familiarity with vector fields and their properties
  • Knowledge of the method of integrating factors
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Given an ODE in the form of f(t)y''+g(t)y'+h(t)y=0

If all I am looking for is the y(t) at a specific value of t and NOT the general solution, can I just plug in that value of t into the original ODE and then solve it analytically or is a numeric solution the only way?
 
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Plugging in a value of t would be a very coarse approximation near that value of t. Not guaranteed to be accurate. The only way you might get away with it is if you had some kind of error bounds.

Plus, you might have an initial condition that is far away from that value of t. In that case, it's likely to be way off.

Differential equations are basically just vector fields (time-varying vector fields, in this case). A solution to a differential equation is obtained by just going where the arrows point. With that intuition in mind, it should be clear what is wrong with your proposed method. If you are following the arrows as they vary in time, there's no reason that picking some point in time will work over long periods of time. The vector field may be completely different at another point in time, so it will take you in a very different direction.

I think maybe the method of integrating factors might work here.
 

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