68.17.4 problem 4

Internal problem ID [17822]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.9, page 215
Problem number : 4
Date solved : Thursday, October 02, 2025 at 02:28:33 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}-25\right )^{2} y^{\prime \prime }-\left (5+x \right ) y^{\prime }+10 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 59

Order:=6; 
ode:=(x^2-25)^2*diff(diff(y(x),x),x)-(x+5)*diff(y(x),x)+10*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {1}{125} x^{2}-\frac {1}{46875} x^{3}-\frac {767}{7812500} x^{4}-\frac {4813}{7324218750} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{250} x^{2}-\frac {112}{46875} x^{3}+\frac {173}{3906250} x^{4}-\frac {409681}{7324218750} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 70

ode=(x^2-25)^2*D[y[x],{x,2}]-(x+5)*D[y[x],x]+10*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (-\frac {4813 x^5}{7324218750}-\frac {767 x^4}{7812500}-\frac {x^3}{46875}-\frac {x^2}{125}+1\right )+c_2 \left (-\frac {409681 x^5}{7324218750}+\frac {173 x^4}{3906250}-\frac {112 x^3}{46875}+\frac {x^2}{250}+x\right ) \]

✓ Sympy. Time used: 0.397 (sec). Leaf size: 68

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-x - 5)*Derivative(y(x), x) + (x**2 - 25)**2*Derivative(y(x), (x, 2)) + 10*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = \frac {x^{4} r{\left (3 \right )}}{500} + \frac {3663 x^{5} r{\left (3 \right )}}{156250} + C_{2} \left (- \frac {23 x^{5}}{146484375} - \frac {23 x^{4}}{234375} - \frac {x^{2}}{125} + 1\right ) + C_{1} x \left (\frac {23 x^{4}}{292968750} + \frac {23 x^{3}}{468750} + \frac {x}{250} + 1\right ) + O\left (x^{6}\right ) \]