Internal problem ID [5691]
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(a).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+y \sin \left (x \right )=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 49
Order:=8; dsolve(diff(y(x),x$2)+sin(x)*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = \left (1-\frac {1}{6} x^{3}+\frac {1}{120} x^{5}+\frac {1}{180} x^{6}-\frac {1}{5040} x^{7}\right ) y \left (0\right )+\left (x -\frac {1}{12} x^{4}+\frac {1}{180} x^{6}+\frac {1}{504} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 63
AsymptoticDSolveValue[y''[x]+Sin[x]*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_2 \left (\frac {x^7}{504}+\frac {x^6}{180}-\frac {x^4}{12}+x\right )+c_1 \left (-\frac {x^7}{5040}+\frac {x^6}{180}+\frac {x^5}{120}-\frac {x^3}{6}+1\right ) \]