20.25.20 problem 21

Internal problem ID [4025]
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 : 21
Date solved : Monday, January 27, 2025 at 08:06:28 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 38

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

\[ y \left (x \right ) = \frac {\left (2 x +x^{2}+\frac {4}{9} x^{3}+\frac {1}{6} x^{4}+\frac {4}{75} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} +\left (1+\operatorname {O}\left (x^{6}\right )\right ) \left (c_{2} \ln \left (x \right )+c_{1} \right )}{x} \]

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 52

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

\[ y(x)\to c_2 \left (\frac {\frac {4 x^5}{75}+\frac {x^4}{6}+\frac {4 x^3}{9}+x^2+2 x}{x}+\frac {\log (x)}{x}\right )+\frac {c_1}{x} \]