68.17.11 problem 11

Internal problem ID [17829]
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 : 11
Date solved : Thursday, October 02, 2025 at 02:28:39 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\frac {8 y^{\prime }}{3 x}-\left (\frac {2}{3 x^{2}}-1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 36

Order:=6; 
ode:=diff(diff(y(x),x),x)+8/3*diff(y(x),x)/x-(2/3/x^2-1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_2 \,x^{{7}/{3}} \left (1-\frac {3}{26} x^{2}+\frac {9}{1976} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_1 \left (1+\frac {3}{2} x^{2}-\frac {9}{40} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{2}} \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 50

ode=D[y[x],{x,2}]+8/3*1/x*D[y[x],x]-(2/3*1/x^2-1)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt [3]{x} \left (\frac {9 x^4}{1976}-\frac {3 x^2}{26}+1\right )+\frac {c_2 \left (-\frac {9 x^4}{40}+\frac {3 x^2}{2}+1\right )}{x^2} \]

✓ Sympy. Time used: 0.451 (sec). Leaf size: 54

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((1 - 2/(3*x**2))*y(x) + Derivative(y(x), (x, 2)) + 8*Derivative(y(x), x)/(3*x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \sqrt [3]{x} \left (\frac {9 x^{4}}{1976} - \frac {3 x^{2}}{26} + 1\right ) + \frac {C_{1} \left (\frac {9 x^{6}}{880} - \frac {9 x^{4}}{40} + \frac {3 x^{2}}{2} + 1\right )}{x^{2}} + O\left (x^{6}\right ) \]