Internal problem ID [6468]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven
Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 4. Power Series Solutions and Special Functions. Section 4.5. More on Regular
Singular Points. Page 183
Problem number: 6.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Bessel]
\[ \boxed {x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 53
Order:=8; dsolve(x^2*diff(y(x),x$2)+x*diff(y(x),x)+(x^2-1)*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = \frac {c_{1} x^{2} \left (1-\frac {1}{8} x^{2}+\frac {1}{192} x^{4}-\frac {1}{9216} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (\left (x^{2}-\frac {1}{8} x^{4}+\frac {1}{192} x^{6}+\operatorname {O}\left (x^{8}\right )\right ) \ln \left (x \right )+\left (-2+\frac {3}{32} x^{4}-\frac {7}{1152} x^{6}+\operatorname {O}\left (x^{8}\right )\right )\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.013 (sec). Leaf size: 75
AsymptoticDSolveValue[x^2*y''[x]+x*y'[x]+(x^2-1)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_2 \left (-\frac {x^7}{9216}+\frac {x^5}{192}-\frac {x^3}{8}+x\right )+c_1 \left (\frac {5 x^6-90 x^4+288 x^2+1152}{1152 x}-\frac {1}{384} x \left (x^4-24 x^2+192\right ) \log (x)\right ) \]