20.26.32 problem 26

Internal problem ID [4057]
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 11.5. page 771
Problem number : 26
Date solved : Monday, January 27, 2025 at 08:07:13 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }+4 x \left (2 x +1\right ) y^{\prime }+\left (4 x -1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.017 (sec). Leaf size: 46

Order:=6; 
dsolve(4*x^2*diff(y(x),x$2)+4*x*(1+2*x)*diff(y(x),x)+(4*x-1)*y(x)=0,y(x),type='series',x=0);
 

\[ y \left (x \right ) = \frac {c_{1} x \left (1-x +\frac {2}{3} x^{2}-\frac {1}{3} x^{3}+\frac {2}{15} x^{4}-\frac {2}{45} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1-2 x +2 x^{2}-\frac {4}{3} x^{3}+\frac {2}{3} x^{4}-\frac {4}{15} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}} \]

✓ Solution by Mathematica

Time used: 0.026 (sec). Leaf size: 88

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

\[ y(x)\to c_1 \left (\frac {2 x^{7/2}}{3}-\frac {4 x^{5/2}}{3}+2 x^{3/2}-2 \sqrt {x}+\frac {1}{\sqrt {x}}\right )+c_2 \left (\frac {2 x^{9/2}}{15}-\frac {x^{7/2}}{3}+\frac {2 x^{5/2}}{3}-x^{3/2}+\sqrt {x}\right ) \]