Internal problem ID [13673]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 35. Modified Power series solutions and basic method of Frobenius. Additional
Exercises. page 715
Problem number: 35.4 (g).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Bessel]
\[ \boxed {y^{\prime \prime }+\frac {y^{\prime }}{x}+\left (1-\frac {1}{x^{2}}\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 47
Order:=6; dsolve(diff(y(x),x$2)+1/x*diff(y(x),x)+(1-1/x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = \frac {c_{1} x^{2} \left (1-\frac {1}{8} x^{2}+\frac {1}{192} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (x^{2}-\frac {1}{8} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2+\frac {3}{32} x^{4}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.01 (sec). Leaf size: 58
AsymptoticDSolveValue[y''[x]+1/x*y'[x]+(1-1/x^2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {x^5}{192}-\frac {x^3}{8}+x\right )+c_1 \left (\frac {1}{16} x \left (x^2-8\right ) \log (x)-\frac {5 x^4-16 x^2-64}{64 x}\right ) \]