36.6.5 problem 5
Internal
problem
ID
[6366]
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
Date
solved
:
Monday, January 27, 2025 at 01:58:14 PM
CAS
classification
:
[_Lienard]
\begin{align*} y^{\prime \prime }-\tan \left (x \right ) y^{\prime }+y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 1 \end{align*}
✓ Solution by Maple
Time used: 0.007 (sec). Leaf size: 125
Order:=6;
dsolve(diff(y(x),x$2)-tan(x)*diff(y(x),x)+y(x)=0,y(x),type='series',x=1);
\[
y = \left (1-\frac {\left (x -1\right )^{2}}{2}-\frac {\tan \left (1\right ) \left (x -1\right )^{3}}{6}+\left (\frac {1}{12}-\frac {\sec \left (1\right )^{2}}{8}\right ) \left (x -1\right )^{4}+\frac {\tan \left (1\right ) \left (1-4 \sec \left (1\right )^{2}\right ) \left (x -1\right )^{5}}{40}\right ) y \left (1\right )+\left (x -1+\frac {\tan \left (1\right ) \left (x -1\right )^{2}}{2}+\frac {\tan \left (1\right )^{2} \left (x -1\right )^{3}}{3}+\frac {\tan \left (1\right ) \left (-1+2 \sec \left (1\right )^{2}\right ) \left (x -1\right )^{4}}{8}+\frac {\left (5-27 \sec \left (1\right )^{2}+24 \sec \left (1\right )^{4}\right ) \left (x -1\right )^{5}}{120}\right ) y^{\prime }\left (1\right )+O\left (x^{6}\right )
\]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 442
AsymptoticDSolveValue[D[y[x],{x,2}]-Tan[x]*D[y[x],x]+y[x]==0,y[x],{x,1,"6"-1}]
\[
y(x)\to c_1 \left (\frac {1}{24} (x-1)^4-\frac {1}{2} (x-1)^2+\frac {1}{20} (x-1)^5 \left (-\tan ^3(1)-\tan (1)\right )-\frac {1}{120} (x-1)^5 \tan ^3(1)-\frac {1}{40} (x-1)^5 \tan (1) \left (1+\tan ^2(1)\right )+\frac {1}{60} (x-1)^5 \tan (1) \left (-1-\tan ^2(1)\right )+\frac {1}{12} (x-1)^4 \left (-1-\tan ^2(1)\right )-\frac {1}{24} (x-1)^4 \tan ^2(1)+\frac {1}{60} (x-1)^5 \tan (1)-\frac {1}{6} (x-1)^3 \tan (1)+1\right )+c_2 \left (\frac {1}{120} (x-1)^5-\frac {1}{6} (x-1)^3+x+\frac {1}{120} (x-1)^5 \tan ^4(1)-\frac {1}{15} (x-1)^5 \tan (1) \left (-\tan ^3(1)-\tan (1)\right )-\frac {1}{12} (x-1)^4 \left (-\tan ^3(1)-\tan (1)\right )+\frac {1}{24} (x-1)^4 \tan ^3(1)-\frac {1}{40} (x-1)^5 \left (-1-\tan ^2(1)\right ) \left (1+\tan ^2(1)\right )+\frac {1}{40} (x-1)^5 \tan ^2(1) \left (1+\tan ^2(1)\right )-\frac {1}{40} (x-1)^5 \left (1+\tan ^2(1)\right )-\frac {1}{40} (x-1)^5 \tan ^2(1) \left (-1-\tan ^2(1)\right )+\frac {1}{120} (x-1)^5 \left (-1-\tan ^2(1)\right )-\frac {1}{40} (x-1)^5 \tan ^2(1)-\frac {1}{8} (x-1)^4 \tan (1) \left (-1-\tan ^2(1)\right )-\frac {1}{6} (x-1)^3 \left (-1-\tan ^2(1)\right )+\frac {1}{6} (x-1)^3 \tan ^2(1)-\frac {1}{60} (x-1)^5 \left (-1-3 \tan ^4(1)-4 \tan ^2(1)\right )-\frac {1}{12} (x-1)^4 \tan (1)+\frac {1}{2} (x-1)^2 \tan (1)-1\right )
\]