(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

This is a problem from an old exam I am reviewing for practice.

Find a good approximation, for x large and positive to the solution of the following equation:

y''-(3/x)y'+(15/(4x^2)+x^(1/2))y=0

Hint: remove first derivative term

2. Relevant equations

3. The attempt at a solutionI'm not sure first of all that I understand the logic behind the hint. After taking this suggestion, my first inclination is to throw away the 15/4x^2 term, leaving y''+sqrt(x)y=0. This is an equation that while it seems simple, I'm ashamed to say I can't figure out how to solve.

Also, according to the problem's solution, you actually throw away just the x^(1/2)y term. This leaves an easily solvable homogeneous Cauchy-Euler DE. But the logic doesn't seem to make sense to me. The solution says that x^(1/2)y and (3/x)y' are negligible at large x. However, isn't the 15/4x^2 term much smaller than these terms?

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# Approximate solution to DE - confused

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