74.17.11 problem 11

Internal problem ID [16542]
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 : Tuesday, January 28, 2025 at 09:10:29 AM
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*}

✓ Solution by Maple

Time used: 0.046 (sec). Leaf size: 36

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

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

✓ Solution by Mathematica

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

AsymptoticDSolveValue[D[y[x],{x,2}]+8/3*1/x*D[y[x],x]-(2/3*1/x^2-1)*y[x]==0,y[x],{x,0,"6"-1}]
 

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