Internal problem ID [4534]
Book: Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson
2018.
Section: Chapter 8, Series solutions of differential equations. Section 8.4. page 449
Problem number: 27.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime }+y-\tan \relax (x )=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 53
Order:=6; dsolve((1-x^2)*diff(y(x),x$2)-diff(y(x),x)+y(x)=tan(x),y(x),type='series',x=0);
\[ y \relax (x ) = \left (1-\frac {1}{2} x^{2}-\frac {1}{6} x^{3}-\frac {1}{12} x^{4}-\frac {7}{120} x^{5}\right ) y \relax (0)+\left (x +\frac {1}{2} x^{2}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}\right ) D\relax (y )\relax (0)+\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{15}+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.025 (sec). Leaf size: 197
AsymptoticDSolveValue[(1-x^2)*y''[x]-y'[x]+y[x]==Tan[x],y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {x^6}{60}+\frac {x^5}{120}+\frac {x^4}{24}+\frac {x^2}{2}+x\right )+c_1 \left (-\frac {7 x^5}{120}-\frac {x^4}{12}-\frac {x^3}{6}-\frac {x^2}{2}+1\right )+\left (-\frac {7 x^5}{120}-\frac {x^4}{12}-\frac {x^3}{6}-\frac {x^2}{2}+1\right ) \left (\frac {7 x^6}{48}-\frac {4 x^5}{15}+\frac {x^4}{8}-\frac {x^3}{3}\right )+\left (\frac {x^6}{60}+\frac {x^5}{120}+\frac {x^4}{24}+\frac {x^2}{2}+x\right ) \left (\frac {67 x^6}{240}-\frac {3 x^5}{10}+\frac {x^4}{3}-\frac {x^3}{3}+\frac {x^2}{2}\right ) \]