45.1.2 problem 15 (x=1)

Internal problem ID [7202]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS with Modeling Applications. Dennis G. Zill. 9th edition. Brooks/Cole. CA, USA.
Section : Chapter 6. SERIES SOLUTIONS OF LINEAR EQUATIONS. Exercises. 6.1.2 page 230
Problem number : 15 (x=1)
Date solved : Sunday, March 30, 2025 at 11:51:13 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}-25\right ) y^{\prime \prime }+2 x y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 1 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 76

Order:=6; 
ode:=(x^2-25)*diff(diff(y(x),x),x)+2*x*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x),type='series',x=1);
 

\[ y = \left (1+\frac {\left (x -1\right )^{2}}{48}+\frac {\left (x -1\right )^{3}}{864}+\frac {\left (x -1\right )^{4}}{1728}+\frac {29 \left (x -1\right )^{5}}{414720}\right ) y \left (1\right )+\left (x -1+\frac {\left (x -1\right )^{2}}{24}+\frac {5 \left (x -1\right )^{3}}{216}+\frac {17 \left (x -1\right )^{4}}{6912}+\frac {41 \left (x -1\right )^{5}}{51840}\right ) y^{\prime }\left (1\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 87

ode=(x^2-25)*D[y[x],{x,2}]+2*x*D[y[x],x]+y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,1,5}]
 

\[ y(x)\to c_1 \left (\frac {29 (x-1)^5}{414720}+\frac {(x-1)^4}{1728}+\frac {1}{864} (x-1)^3+\frac {1}{48} (x-1)^2+1\right )+c_2 \left (\frac {41 (x-1)^5}{51840}+\frac {17 (x-1)^4}{6912}+\frac {5}{216} (x-1)^3+\frac {1}{24} (x-1)^2+x-1\right ) \]

✓ Sympy. Time used: 0.991 (sec). Leaf size: 58

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x*Derivative(y(x), x) + (x**2 - 25)*Derivative(y(x), (x, 2)) + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=1,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (x + \frac {17 \left (x - 1\right )^{4}}{6912} + \frac {5 \left (x - 1\right )^{3}}{216} + \frac {\left (x - 1\right )^{2}}{24} - 1\right ) + C_{1} \left (\frac {\left (x - 1\right )^{4}}{1728} + \frac {\left (x - 1\right )^{3}}{864} + \frac {\left (x - 1\right )^{2}}{48} + 1\right ) + O\left (x^{6}\right ) \]