52.4.5 problem 13

Internal problem ID [8315]
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. CHAPTER 6 IN REVIEW. Page 271
Problem number : 13
Date solved : Monday, January 27, 2025 at 03:48:44 PM
CAS classification : [_Laguerre]

\begin{align*} x y^{\prime \prime }-\left (x +2\right ) y^{\prime }+2 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: 52

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

\[ y = c_{1} x^{3} \left (1+\frac {1}{4} x +\frac {1}{20} x^{2}+\frac {1}{120} x^{3}+\frac {1}{840} x^{4}+\frac {1}{6720} x^{5}+\frac {1}{60480} x^{6}+\frac {1}{604800} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (12+12 x +6 x^{2}+2 x^{3}+\frac {1}{2} x^{4}+\frac {1}{10} x^{5}+\frac {1}{60} x^{6}+\frac {1}{420} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.088 (sec). Leaf size: 94

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

\[ y(x)\to c_1 \left (\frac {x^6}{720}+\frac {x^5}{120}+\frac {x^4}{24}+\frac {x^3}{6}+\frac {x^2}{2}+x+1\right )+c_2 \left (\frac {x^9}{60480}+\frac {x^8}{6720}+\frac {x^7}{840}+\frac {x^6}{120}+\frac {x^5}{20}+\frac {x^4}{4}+x^3\right ) \]