Internal problem ID [4515]
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: 5.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Lienard]
Solve \begin {gather*} \boxed {y^{\prime \prime }-\tan \relax (x ) y^{\prime }+y=0} \end {gather*} With the expansion point for the power series method at \(x = 1\).
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 106
Order:=6; dsolve(diff(y(x),x$2)-tan(x)*diff(y(x),x)+y(x)=0,y(x),type='series',x=1);
\[ y \relax (x ) = \left (1-\frac {\left (x -1\right )^{2}}{2}-\frac {\tan \relax (1) \left (x -1\right )^{3}}{6}+\left (-\frac {1}{24}-\frac {\left (\tan ^{2}\relax (1)\right )}{8}\right ) \left (x -1\right )^{4}+\left (-\frac {3 \tan \relax (1)}{40}-\frac {\left (\tan ^{3}\relax (1)\right )}{10}\right ) \left (x -1\right )^{5}\right ) y \relax (1)+\left (x -1+\frac {\tan \relax (1) \left (x -1\right )^{2}}{2}+\frac {\left (\tan ^{2}\relax (1)\right ) \left (x -1\right )^{3}}{3}+\left (\frac {\tan \relax (1)}{8}+\frac {\left (\tan ^{3}\relax (1)\right )}{4}\right ) \left (x -1\right )^{4}+\left (\frac {7 \left (\tan ^{2}\relax (1)\right )}{40}+\frac {\left (\tan ^{4}\relax (1)\right )}{5}+\frac {1}{60}\right ) \left (x -1\right )^{5}\right ) D\relax (y )\relax (1)+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 442
AsymptoticDSolveValue[y''[x]-Tan[x]*y'[x]+y[x]==0,y[x],{x,1,5}]
\[ y(x)\to c_1 \left (\frac {1}{24} (x-1)^4-\frac {1}{2} (x-1)^2+\frac {1}{20} (x-1)^5 \left (-\tan ^3(1)-\tan (1)\right )-\frac {1}{120} (x-1)^5 \tan ^3(1)-\frac {1}{40} (x-1)^5 \tan (1) \left (1+\tan ^2(1)\right )+\frac {1}{60} (x-1)^5 \tan (1) \left (-1-\tan ^2(1)\right )+\frac {1}{12} (x-1)^4 \left (-1-\tan ^2(1)\right )-\frac {1}{24} (x-1)^4 \tan ^2(1)+\frac {1}{60} (x-1)^5 \tan (1)-\frac {1}{6} (x-1)^3 \tan (1)+1\right )+c_2 \left (\frac {1}{120} (x-1)^5-\frac {1}{6} (x-1)^3+x+\frac {1}{120} (x-1)^5 \tan ^4(1)-\frac {1}{15} (x-1)^5 \tan (1) \left (-\tan ^3(1)-\tan (1)\right )-\frac {1}{12} (x-1)^4 \left (-\tan ^3(1)-\tan (1)\right )+\frac {1}{24} (x-1)^4 \tan ^3(1)-\frac {1}{40} (x-1)^5 \left (-1-\tan ^2(1)\right ) \left (1+\tan ^2(1)\right )+\frac {1}{40} (x-1)^5 \tan ^2(1) \left (1+\tan ^2(1)\right )-\frac {1}{40} (x-1)^5 \left (1+\tan ^2(1)\right )-\frac {1}{40} (x-1)^5 \tan ^2(1) \left (-1-\tan ^2(1)\right )+\frac {1}{120} (x-1)^5 \left (-1-\tan ^2(1)\right )-\frac {1}{40} (x-1)^5 \tan ^2(1)-\frac {1}{8} (x-1)^4 \tan (1) \left (-1-\tan ^2(1)\right )-\frac {1}{6} (x-1)^3 \left (-1-\tan ^2(1)\right )+\frac {1}{6} (x-1)^3 \tan ^2(1)-\frac {1}{60} (x-1)^5 \left (-1-3 \tan ^4(1)-4 \tan ^2(1)\right )-\frac {1}{12} (x-1)^4 \tan (1)+\frac {1}{2} (x-1)^2 \tan (1)-1\right ) \]