78.22.2 problem 1 (b)

Internal problem ID [18451]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 5. Power Series Solutions and Special Functions. Section 32. The point at infinity. Problems at page 244
Problem number : 1 (b)
Date solved : Tuesday, January 28, 2025 at 11:50:17 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*} \infty \end{align*}

✗ Solution by Maple

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=Infinity);
 

\[ \text {No solution found} \]

✓ Solution by Mathematica

Time used: 0.050 (sec). Leaf size: 534

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

\[ y(x)\to c_1 e^{-i x} \left (-\frac {i n^{10}}{3840 x^{11/2}}+\frac {n^8}{384 x^{9/2}}+\frac {11 i n^8}{1024 x^{11/2}}+\frac {i n^6}{48 x^{7/2}}-\frac {7 n^6}{128 x^{9/2}}-\frac {1463 i n^6}{10240 x^{11/2}}-\frac {n^4}{8 x^{5/2}}-\frac {35 i n^4}{192 x^{7/2}}+\frac {329 n^4}{1024 x^{9/2}}+\frac {17281 i n^4}{24576 x^{11/2}}-\frac {i n^2}{2 x^{3/2}}+\frac {5 n^2}{16 x^{5/2}}+\frac {259 i n^2}{768 x^{7/2}}-\frac {3229 n^2}{6144 x^{9/2}}-\frac {352407 i n^2}{327680 x^{11/2}}+\frac {i}{8 x^{3/2}}-\frac {9}{128 x^{5/2}}-\frac {75 i}{1024 x^{7/2}}+\frac {3675}{32768 x^{9/2}}+\frac {59535 i}{262144 x^{11/2}}+\frac {1}{\sqrt {x}}\right )+c_2 e^{i x} \left (\frac {i n^{10}}{3840 x^{11/2}}+\frac {n^8}{384 x^{9/2}}-\frac {11 i n^8}{1024 x^{11/2}}-\frac {i n^6}{48 x^{7/2}}-\frac {7 n^6}{128 x^{9/2}}+\frac {1463 i n^6}{10240 x^{11/2}}-\frac {n^4}{8 x^{5/2}}+\frac {35 i n^4}{192 x^{7/2}}+\frac {329 n^4}{1024 x^{9/2}}-\frac {17281 i n^4}{24576 x^{11/2}}+\frac {i n^2}{2 x^{3/2}}+\frac {5 n^2}{16 x^{5/2}}-\frac {259 i n^2}{768 x^{7/2}}-\frac {3229 n^2}{6144 x^{9/2}}+\frac {352407 i n^2}{327680 x^{11/2}}-\frac {i}{8 x^{3/2}}-\frac {9}{128 x^{5/2}}+\frac {75 i}{1024 x^{7/2}}+\frac {3675}{32768 x^{9/2}}-\frac {59535 i}{262144 x^{11/2}}+\frac {1}{\sqrt {x}}\right ) \]