20.26.12 problem 4

Internal problem ID [4037]
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 : 4
Date solved : Monday, January 27, 2025 at 08:06:45 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x^{2} 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: 0.018 (sec). Leaf size: 47

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

\[ y \left (x \right ) = c_{1} x^{2} \left (1-\frac {1}{4} x +\frac {1}{20} x^{2}-\frac {1}{120} x^{3}+\frac {1}{840} x^{4}-\frac {1}{6720} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (12-12 x +6 x^{2}-2 x^{3}+\frac {1}{2} x^{4}-\frac {1}{10} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Solution by Mathematica

Time used: 0.023 (sec). Leaf size: 66

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

\[ y(x)\to c_1 \left (\frac {x^3}{24}-\frac {x^2}{6}+\frac {x}{2}+\frac {1}{x}-1\right )+c_2 \left (\frac {x^6}{840}-\frac {x^5}{120}+\frac {x^4}{20}-\frac {x^3}{4}+x^2\right ) \]