30.14.5 problem 5
Internal
problem
ID
[7654]
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
:
Tuesday, September 30, 2025 at 04:55:31 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*}
✓ Maple. Time used: 0.014 (sec). Leaf size: 125
Order:=6;
ode:=diff(diff(y(x),x),x)-tan(x)*diff(y(x),x)+y(x) = 0;
dsolve(ode,y(x),type='series',x=1);
\[
y = \left (1-\frac {\left (x -1\right )^{2}}{2}-\frac {\tan \left (1\right ) \left (x -1\right )^{3}}{6}+\left (\frac {1}{12}-\frac {\sec \left (1\right )^{2}}{8}\right ) \left (x -1\right )^{4}+\frac {\tan \left (1\right ) \left (1-4 \sec \left (1\right )^{2}\right ) \left (x -1\right )^{5}}{40}\right ) y \left (1\right )+\left (x -1+\frac {\tan \left (1\right ) \left (x -1\right )^{2}}{2}+\frac {\tan \left (1\right )^{2} \left (x -1\right )^{3}}{3}+\frac {\tan \left (1\right ) \left (-1+2 \sec \left (1\right )^{2}\right ) \left (x -1\right )^{4}}{8}+\frac {\left (5-27 \sec \left (1\right )^{2}+24 \sec \left (1\right )^{4}\right ) \left (x -1\right )^{5}}{120}\right ) y^{\prime }\left (1\right )+O\left (x^{6}\right )
\]
✓ Mathematica. Time used: 0.006 (sec). Leaf size: 289
ode=D[y[x],{x,2}]-Tan[x]*D[y[x],x]+y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,1,5}]
\[
y(x)\to c_1 \left (\frac {1}{24} (x-1)^4-\frac {1}{2} (x-1)^2-\frac {1}{120} (x-1)^5 \tan ^3(1)-\frac {1}{24} (x-1)^4 \tan ^2(1)+\frac {1}{60} (x-1)^5 \tan (1)-\frac {1}{6} (x-1)^3 \tan (1)-\frac {1}{12} (x-1)^4 \sec ^2(1)-\frac {11}{120} (x-1)^5 \tan (1) \sec ^2(1)+1\right )+c_2 \left (\frac {1}{120} (x-1)^5-\frac {1}{6} (x-1)^3+x+\frac {1}{120} (x-1)^5 \tan ^4(1)+\frac {1}{24} (x-1)^4 \tan ^3(1)-\frac {1}{40} (x-1)^5 \tan ^2(1)+\frac {1}{6} (x-1)^3 \tan ^2(1)-\frac {1}{12} (x-1)^4 \tan (1)+\frac {1}{2} (x-1)^2 \tan (1)+\frac {1}{40} (x-1)^5 \sec ^4(1)-\frac {1}{30} (x-1)^5 \sec ^2(1)+\frac {1}{6} (x-1)^3 \sec ^2(1)-\frac {1}{60} (x-1)^5 (\cos (2)-2) \sec ^4(1)+\frac {7}{60} (x-1)^5 \tan ^2(1) \sec ^2(1)+\frac {5}{24} (x-1)^4 \tan (1) \sec ^2(1)-1\right )
\]
✓ Sympy. Time used: 1.273 (sec). Leaf size: 110
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(y(x) - tan(x)*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=1,n=6)
\[
y{\left (x \right )} = C_{2} \left (x + \frac {\left (x - 1\right )^{4} \tan ^{3}{\left (x + 1 \right )}}{24} - \frac {\left (x - 1\right )^{4} \tan {\left (x + 1 \right )}}{12} + \frac {\left (x - 1\right )^{3} \tan ^{2}{\left (x + 1 \right )}}{6} - \frac {\left (x - 1\right )^{3}}{6} + \frac {\left (x - 1\right )^{2} \tan {\left (x + 1 \right )}}{2} - 1\right ) + C_{1} \left (- \frac {\left (x - 1\right )^{4} \tan ^{2}{\left (x + 1 \right )}}{24} + \frac {\left (x - 1\right )^{4}}{24} - \frac {\left (x - 1\right )^{3} \tan {\left (x + 1 \right )}}{6} - \frac {\left (x - 1\right )^{2}}{2} + 1\right ) + O\left (x^{6}\right )
\]