Internal problem ID [5697]
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: 3(a).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{3} y^{\prime \prime }+\left (-1+\cos \left (2 x \right )\right ) y^{\prime }+2 x 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: 37
Order:=8; dsolve(x^3*diff(y(x),x$2)+(cos(2*x)-1)*diff(y(x),x)+2*x*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{2} \left (1-\frac {2}{9} x^{2}+\frac {26}{675} x^{4}-\frac {1742}{297675} x^{6}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} x \left (1-\frac {1}{3} x^{2}+\frac {17}{270} x^{4}-\frac {173}{17010} x^{6}+\mathrm {O}\left (x^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.035 (sec). Leaf size: 74
AsymptoticDSolveValue[x^3*y''[x]+(Cos[2*x]-1)*y'[x]+2*x*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_2 \left (\frac {32351 x^8}{40186125}-\frac {1742 x^6}{297675}+\frac {26 x^4}{675}-\frac {2 x^2}{9}+1\right ) x^2+c_1 \left (\frac {10471 x^8}{7144200}-\frac {173 x^6}{17010}+\frac {17 x^4}{270}-\frac {x^2}{3}+1\right ) x \]