20.26.35 problem 29

Internal problem ID [4060]
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 : 29
Date solved : Monday, January 27, 2025 at 08:07:17 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.019 (sec). Leaf size: 46

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

\[ y \left (x \right ) = \frac {\ln \left (x \right ) \left (x +\operatorname {O}\left (x^{6}\right )\right ) c_{2} +c_{1} x \left (1+\operatorname {O}\left (x^{6}\right )\right )+\left (1-x -\frac {1}{2} x^{2}+\frac {1}{12} x^{3}-\frac {1}{72} x^{4}+\frac {1}{480} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2}}{x^{2}} \]

✓ Solution by Mathematica

Time used: 0.027 (sec). Leaf size: 45

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

\[ y(x)\to c_1 \left (\frac {\log (x)}{x}-\frac {x^4-6 x^3+36 x^2+144 x-72}{72 x^2}\right )+\frac {c_2}{x} \]