Internal problem ID [5694]
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 4.4. REGULAR SINGULAR
POINTS. Page 175
Problem number: 2(c).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }+\sin \relax (x ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.093 (sec). Leaf size: 70
Order:=8; dsolve(x^2*diff(y(x),x$2)+sin(x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x \left (1-\frac {1}{2} x +\frac {1}{12} x^{2}+\frac {1}{144} x^{3}-\frac {13}{2880} x^{4}+\frac {29}{86400} x^{5}+\frac {431}{3628800} x^{6}-\frac {4961}{203212800} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} \left (\ln \relax (x ) \left (-x +\frac {1}{2} x^{2}-\frac {1}{12} x^{3}-\frac {1}{144} x^{4}+\frac {13}{2880} x^{5}-\frac {29}{86400} x^{6}-\frac {431}{3628800} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+\left (1-\frac {3}{4} x^{2}+\frac {2}{9} x^{3}-\frac {25}{1728} x^{4}-\frac {689}{86400} x^{5}+\frac {263}{162000} x^{6}+\frac {71809}{762048000} x^{7}+\mathrm {O}\left (x^{8}\right )\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.066 (sec). Leaf size: 121
AsymptoticDSolveValue[x^2*y''[x]+Sin[x]*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (\frac {2539 x^6-16185 x^5-9750 x^4+396000 x^3-1620000 x^2+1296000 x+1296000}{1296000}-\frac {x \left (29 x^5-390 x^4+600 x^3+7200 x^2-43200 x+86400\right ) \log (x)}{86400}\right )+c_2 \left (\frac {431 x^7}{3628800}+\frac {29 x^6}{86400}-\frac {13 x^5}{2880}+\frac {x^4}{144}+\frac {x^3}{12}-\frac {x^2}{2}+x\right ) \]