Internal problem ID [6272]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 18. Power series solutions. Miscellaneous Exercises. page 394
Problem number: 24.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Bessel, _modified]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }+x y^{\prime }-\left (x^{2}+4\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 53
Order:=8; dsolve(x^2*diff(y(x),x$2)+x*diff(y(x),x)-(x^2+4)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{1} x^{4} \left (1+\frac {1}{12} x^{2}+\frac {1}{384} x^{4}+\frac {1}{23040} x^{6}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} \left (\ln \relax (x ) \left (9 x^{4}+\frac {3}{4} x^{6}+\mathrm {O}\left (x^{8}\right )\right )+\left (-144+36 x^{2}-\frac {1}{2} x^{6}+\mathrm {O}\left (x^{8}\right )\right )\right )}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.021 (sec). Leaf size: 74
AsymptoticDSolveValue[x^2*y''[x]+x*y'[x]-(x^2+4)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (\frac {11 x^6+36 x^4-576 x^2+2304}{2304 x^2}-\frac {1}{192} x^2 \left (x^2+12\right ) \log (x)\right )+c_2 \left (\frac {x^8}{23040}+\frac {x^6}{384}+\frac {x^4}{12}+x^2\right ) \]