Zeroes of ODE in normal formI need to find the number of zeroes of an ODE:yq(x)y=0when q(x)&gt;0 but monotonic decreasing. For instance:yyx2=0where x∈[1,∞).How can I use the Strum-Comparision Theorem to estimate the number of zeroes of it's solution?(P.S. Here the number of zeroes won't be infinitely many contrary to the general case of q(x)&gt;0 and monotonic increasing.)

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Zeroes of ODE in normal formI need to find the number of zeroes of an ODE:yq(x)y=0when q(x)&amp;gt;0 but monotonic decreasing. For instance:yyx2=0where x∈[1,∞).How can I use the Strum-Comparision Theorem to estimate the number of zeroes of it&#039;s solution?(P.S. Here the number of zeroes won&#039;t be infinitely many contrary to the general case of q(x)&amp;gt;0 and monotonic increasing.)

Ian Adams · Accepted Answer

Step 1The example you can solve exactly, as that is Euler-Cauchy,x2y″(x)+y(x)=0⟹y(x)=Ax12cos⁡(32ln⁡(x)+B)giving infinitely many zeros.Step 2In the more general case, you can get a first impression by using the WKB approximationy≈Aq(x)−14cos⁡(B+∫1xq(s)ds)to get a general idea if the number of zeros will be finite or infinite.

Zeroes of ODE in normal form I need to find the number of zeroes of an ODE: y"+q(x)y=0

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