20.26.30 problem 24

Internal problem ID [4055]
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 : 24
Date solved : Monday, January 27, 2025 at 08:07:10 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.021 (sec). Leaf size: 46

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

\[ y \left (x \right ) = \frac {\ln \left (x \right ) \left (2 x +\operatorname {O}\left (x^{6}\right )\right ) c_{2} +c_{1} x \left (1+\operatorname {O}\left (x^{6}\right )\right )+\left (1-2 x -2 x^{2}+\frac {2}{3} x^{3}-\frac {2}{9} x^{4}+\frac {1}{15} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2}}{x^{2}} \]

✓ Solution by Mathematica

Time used: 0.029 (sec). Leaf size: 48

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

\[ y(x)\to c_1 \left (\frac {2 \log (x)}{x}-\frac {2 x^4-6 x^3+18 x^2+36 x-9}{9 x^2}\right )+\frac {c_2}{x} \]