74.17.13 problem 13

Internal problem ID [16544]
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 : 13
Date solved : Tuesday, January 28, 2025 at 09:10:31 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\left (\frac {1}{2 x}-2\right ) y^{\prime }-\frac {35 y}{16 x^{2}}&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.072 (sec). Leaf size: 62

Order:=6; 
dsolve(diff(y(x),x$2)+(1/2*1/x-2)*diff(y(x),x)-35/16*1/x^2*y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {c_{1} x^{3} \left (1+\frac {7}{8} x +\frac {77}{160} x^{2}+\frac {77}{384} x^{3}+\frac {209}{3072} x^{4}+\frac {4807}{245760} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (\frac {15}{8} x^{3}+\frac {105}{64} x^{4}+\frac {231}{256} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (12+15 x +\frac {15}{4} x^{2}-\frac {13}{2} x^{3}-\frac {1741}{256} x^{4}-\frac {4141}{1024} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{{5}/{4}}} \]

✓ Solution by Mathematica

Time used: 0.059 (sec). Leaf size: 98

AsymptoticDSolveValue[D[y[x],{x,2}]+(1/2*1/x-2)*D[y[x],x]-35/16*1/x^2*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_2 \left (\frac {209 x^{23/4}}{3072}+\frac {77 x^{19/4}}{384}+\frac {77 x^{15/4}}{160}+\frac {7 x^{11/4}}{8}+x^{7/4}\right )+c_1 \left (\frac {5}{256} x^{7/4} (7 x+8) \log (x)-\frac {627 x^4+608 x^3-320 x^2-1280 x-1024}{1024 x^{5/4}}\right ) \]