Internal problem ID [5293]
Book: An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY
1961
Section: Chapter 4. Linear equations with Regular Singular Points. Page 154
Problem number: 1(g).
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 }+y^{\prime } \sin \relax (x )+\cos \relax (x ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.156 (sec). Leaf size: 53
Order:=8; dsolve(x^2*diff(y(x),x$2)+sin(x)*diff(y(x),x)+cos(x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{-i} \left (1+\left (\frac {1}{12}+\frac {i}{24}\right ) x^{2}+\left (\frac {29}{28800}+\frac {67 i}{28800}\right ) x^{4}+\left (-\frac {893}{14515200}-\frac {17 i}{4838400}\right ) x^{6}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} x^{i} \left (1+\left (\frac {1}{12}-\frac {i}{24}\right ) x^{2}+\left (\frac {29}{28800}-\frac {67 i}{28800}\right ) x^{4}+\left (-\frac {893}{14515200}+\frac {17 i}{4838400}\right ) x^{6}+\mathrm {O}\left (x^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.075 (sec). Leaf size: 112
AsymptoticDSolveValue[x^2*y''[x]+Sin[x]*y'[x]+Cos[x]*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 x^{-i} \left (\left (-\frac {26459}{59222016000}-\frac {12449 i}{7402752000}\right ) x^8-\left (\frac {893}{14515200}+\frac {17 i}{4838400}\right ) x^6+\left (\frac {29}{28800}+\frac {67 i}{28800}\right ) x^4+\left (\frac {1}{12}+\frac {i}{24}\right ) x^2+1\right )+c_2 x^i \left (\left (-\frac {26459}{59222016000}+\frac {12449 i}{7402752000}\right ) x^8-\left (\frac {893}{14515200}-\frac {17 i}{4838400}\right ) x^6+\left (\frac {29}{28800}-\frac {67 i}{28800}\right ) x^4+\left (\frac {1}{12}-\frac {i}{24}\right ) x^2+1\right ) \]