Internal problem ID [5696]
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(e).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{4} 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
Order:=8; dsolve(x^4*diff(y(x),x$2)+sin(x)*y(x)=0,y(x),type='series',x=0);
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 0.172 (sec). Leaf size: 294
AsymptoticDSolveValue[x^4*y''[x]+Sin[x]*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 e^{-\frac {2 i}{\sqrt {x}}} x^{3/4} \left (\frac {16487484152477478659746223 i x^{13/2}}{2773583263632691770163200}-\frac {4594934148364735183693 i x^{11/2}}{6320013947079701299200}+\frac {12579783586699513 i x^{9/2}}{96185277197844480}-\frac {21896783401 i x^{7/2}}{579820584960}+\frac {856783 i x^{5/2}}{41943040}-\frac {3151 i x^{3/2}}{73728}-\frac {3986263268940827572255963529 x^7}{207094217017907652172185600}+\frac {21730712888356628741772337 x^6}{10920984100553723845017600}-\frac {1500040357444099007 x^5}{5129881450551705600}+\frac {4885269094757 x^4}{74217034874880}-\frac {2835642457 x^3}{108716359680}+\frac {11659 x^2}{524288}+\frac {15 x}{512}-\frac {3 i \sqrt {x}}{16}+1\right )+c_2 e^{\frac {2 i}{\sqrt {x}}} x^{3/4} \left (-\frac {16487484152477478659746223 i x^{13/2}}{2773583263632691770163200}+\frac {4594934148364735183693 i x^{11/2}}{6320013947079701299200}-\frac {12579783586699513 i x^{9/2}}{96185277197844480}+\frac {21896783401 i x^{7/2}}{579820584960}-\frac {856783 i x^{5/2}}{41943040}+\frac {3151 i x^{3/2}}{73728}-\frac {3986263268940827572255963529 x^7}{207094217017907652172185600}+\frac {21730712888356628741772337 x^6}{10920984100553723845017600}-\frac {1500040357444099007 x^5}{5129881450551705600}+\frac {4885269094757 x^4}{74217034874880}-\frac {2835642457 x^3}{108716359680}+\frac {11659 x^2}{524288}+\frac {15 x}{512}+\frac {3 i \sqrt {x}}{16}+1\right ) \]