52.1.16 problem 14

Internal problem ID [8233]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS. Section 6.2 SOLUTIONS ABOUT ORDINARY POINTS. EXERCISES 6.2. Page 246
Problem number : 14
Date solved : Monday, January 27, 2025 at 03:46:34 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 43

Order:=8; 
dsolve((x+2)*diff(y(x),x$2)+x*diff(y(x),x)-y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (1+\frac {1}{4} x^{2}-\frac {1}{24} x^{3}+\frac {1}{480} x^{5}-\frac {1}{1440} x^{6}+\frac {1}{6720} x^{7}\right ) y \left (0\right )+y^{\prime }\left (0\right ) x +O\left (x^{8}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 91

AsymptoticDSolveValue[(x+2)*D[y[x],{x,2}]+x*D[y[x],x]+y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to c_2 \left (\frac {29 x^7}{20160}-\frac {7 x^6}{1440}+\frac {x^5}{240}+\frac {x^4}{24}-\frac {x^3}{6}+x\right )+c_1 \left (\frac {x^7}{8064}+\frac {x^6}{576}-\frac {x^5}{96}+\frac {x^4}{48}+\frac {x^3}{24}-\frac {x^2}{4}+1\right ) \]