Internal problem ID [6237]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 18. Power series solutions. 18.9 Indicial Equation with Difference of Roots a
Positive Integer: Logarithmic Case. Exercises page 384
Problem number: 10 (as series).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Bessel]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-1\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.029 (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);
\[ y \relax (x ) = \frac {c_{1} x^{2} \left (1-\frac {1}{8} x^{2}+\frac {1}{192} x^{4}-\frac {1}{9216} x^{6}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} \left (\left (x^{2}-\frac {1}{8} x^{4}+\frac {1}{192} x^{6}+\mathrm {O}\left (x^{8}\right )\right ) \ln \relax (x )+\left (-2+\frac {3}{32} x^{4}-\frac {7}{1152} x^{6}+\mathrm {O}\left (x^{8}\right )\right )\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.011 (sec). Leaf size: 75
AsymptoticDSolveValue[x^2*y''[x]+x*y'[x]+(x^2-1)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_2 \left (-\frac {x^7}{9216}+\frac {x^5}{192}-\frac {x^3}{8}+x\right )+c_1 \left (\frac {5 x^6-90 x^4+288 x^2+1152}{1152 x}-\frac {1}{384} x \left (x^4-24 x^2+192\right ) \log (x)\right ) \]