20.25.7 problem 8

Internal problem ID [4012]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.4. page 758
Problem number : 8
Date solved : Monday, January 27, 2025 at 08:06:12 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x y^{\prime \prime }-x y^{\prime }+2 y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.014 (sec). Leaf size: 46

Order:=6; 
dsolve(4*x*diff(y(x),x$2)-x*diff(y(x),x)+2*y(x)=0,y(x),type='series',x=0);
 

\[ y \left (x \right ) = \ln \left (x \right ) \left (-\frac {1}{2} x +\frac {1}{16} x^{2}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} +c_{1} x \left (1-\frac {1}{8} x +\operatorname {O}\left (x^{6}\right )\right )+\left (1+\frac {1}{4} x -\frac {3}{16} x^{2}+\frac {1}{384} x^{3}+\frac {1}{18432} x^{4}+\frac {1}{737280} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

✓ Solution by Mathematica

Time used: 0.027 (sec). Leaf size: 52

AsymptoticDSolveValue[4*x*D[y[x],{x,2}]-x*D[y[x],x]+2*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_2 \left (x-\frac {x^2}{8}\right )+c_1 \left (\frac {x^4+48 x^3-4608 x^2+13824 x+18432}{18432}+\frac {1}{16} (x-8) x \log (x)\right ) \]