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CSS Applied Mathematics QUESTION #1302
Question 1
For the power series solution of \((x^2+1)y''+xy'-y=0\), we assume \(y=\sum_{n=0}^\infty a_n x^n\). This ODE has an ordinary point at \(x=0\) because:
  • The coefficients are all constants
  • All singular points are at \(x=\pm i\), not at \(x=0\)✔️
  • The equation is linear
  • The ODE is homogeneous
Correct Answer Explanation
Ordinary points are where \(P(x)=1+x^2\neq0\). The singular points are where \(P(x)=0\Rightarrow x^2+1=0\Rightarrow x=\pm i\). Since \(x=0\) is not a singular point, it is an ordinary point, and a power series solution \(\sum a_n x^n\) converges in a neighbourhood of \(x=0\).