Internal problem ID [11760]
Book: AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C.
ROBINSON. Cambridge University Press 2004
Section: Chapter 20, Series solutions of second order linear equations. Exercises page
195
Problem number: 20.7.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Bessel]
\[ \boxed {x^{2} y^{\prime \prime }+x y^{\prime }+\left (-n^{2}+x^{2}\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 77
Order:=6; dsolve(x^2*diff(y(x),x$2)+x*diff(y(x),x)+(x^2-n^2)*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = c_{1} x^{-n} \left (1+\frac {1}{4 n -4} x^{2}+\frac {1}{32} \frac {1}{\left (n -2\right ) \left (n -1\right )} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{n} \left (1-\frac {1}{4 n +4} x^{2}+\frac {1}{32} \frac {1}{\left (n +2\right ) \left (n +1\right )} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 160
AsymptoticDSolveValue[x^2*y''[x]+x*y'[x]+(x^2-n^2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {x^4}{\left (-n^2-n+(1-n) (2-n)+2\right ) \left (-n^2-n+(3-n) (4-n)+4\right )}-\frac {x^2}{-n^2-n+(1-n) (2-n)+2}+1\right ) x^{-n}+c_1 \left (\frac {x^4}{\left (-n^2+n+(n+1) (n+2)+2\right ) \left (-n^2+n+(n+3) (n+4)+4\right )}-\frac {x^2}{-n^2+n+(n+1) (n+2)+2}+1\right ) x^n \]