65.13.10 problem 20.7

Internal problem ID [13816]
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
Date solved : Tuesday, January 28, 2025 at 06:04:39 AM
CAS classification : [_Bessel]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+\left (-n^{2}+x^{2}\right ) y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.045 (sec). Leaf size: 73

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);
 

\[ y = c_{1} x^{-n} \left (1+\frac {1}{4 n -4} x^{2}+\frac {1}{32} \frac {1}{\left (n -2\right ) \left (n -1\right )} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{n} \left (1-\frac {1}{4 n +4} x^{2}+\frac {1}{32} \frac {1}{\left (n +2\right ) \left (n +1\right )} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.008 (sec). Leaf size: 160

AsymptoticDSolveValue[x^2*D[y[x],{x,2}]+x*D[y[x],x]+(x^2-n^2)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_2 \left (\frac {x^4}{\left (-n^2-n+(1-n) (2-n)+2\right ) \left (-n^2-n+(3-n) (4-n)+4\right )}-\frac {x^2}{-n^2-n+(1-n) (2-n)+2}+1\right ) x^{-n}+c_1 \left (\frac {x^4}{\left (-n^2+n+(n+1) (n+2)+2\right ) \left (-n^2+n+(n+3) (n+4)+4\right )}-\frac {x^2}{-n^2+n+(n+1) (n+2)+2}+1\right ) x^n \]