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I am dealing with the following problem: consider the ODE $$xy_M''(x) + y_M(x)=0,$$ with boundary data $y(1)=1$ and $y(M)=0$.

I would like to know if letting $M\to \infty$ the sequence $\{y_M\}_M$ converges to the constant function $1$ or to a non-constant solution of the same problem with boundary data $y(1)=1$ and $y(\infty)=0$.

I thank you in advance, Mattia

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    Did you find the solution of the ODE ? – 2017-02-17
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    I think no solutions in elementary functions is available for this problem.If something is available I would be happy! – 2017-02-17
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    The question is (to me !) : what is the limit of elementary functions ? In fact, the ODE makes me thinking about Bessel functions with argument $x\sqrt 2$. – 2017-02-17
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    Ah ok, we can solve it by Bessel functions! But I would like to know if someone would be able to understand the behaviour of the sequence just by a qualitative study of the ODE. – 2017-02-17

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