52.4.2 problem 10

Internal problem ID [8312]
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 : 10
Date solved : Monday, January 27, 2025 at 03:48:41 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} 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: 0.011 (sec). Leaf size: 49

Order:=8; 
dsolve(diff(y(x),x$2)-x*diff(y(x),x)-y(x)=0,y(x),type='series',x=0);
 
\[ y = \left (1+\frac {1}{2} x^{2}+\frac {1}{8} x^{4}+\frac {1}{48} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{3} x^{3}+\frac {1}{15} x^{5}+\frac {1}{105} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 56

AsymptoticDSolveValue[D[y[x],{x,2}]-x*D[y[x],x]-y[x]==0,y[x],{x,0,"8"-1}]
 
\[ y(x)\to c_2 \left (\frac {x^7}{105}+\frac {x^5}{15}+\frac {x^3}{3}+x\right )+c_1 \left (\frac {x^6}{48}+\frac {x^4}{8}+\frac {x^2}{2}+1\right ) \]