52.4.10 problem 18

Internal problem ID [8320]
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 : 18
Date solved : Monday, January 27, 2025 at 03:48:54 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*} 1 \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=-6\\ y^{\prime }\left (1\right )&=3 \end{align*}

✓ Solution by Maple

Time used: 0.000 (sec). Leaf size: 22

Order:=8; 
dsolve([diff(y(x),x$2)+x*diff(y(x),x)+y(x)=0,y(1) = -6, D(y)(1) = 3],y(x),type='series',x=1);
 

\[ y = -6+3 \left (x -1\right )+\frac {3}{2} \left (x -1\right )^{2}-\frac {3}{2} \left (x -1\right )^{3}+\frac {3}{10} \left (x -1\right )^{5}-\frac {1}{20} \left (x -1\right )^{6}-\frac {1}{28} \left (x -1\right )^{7}+\operatorname {O}\left (\left (x -1\right )^{8}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 55

AsymptoticDSolveValue[{D[y[x],{x,2}]+x*D[y[x],x]+y[x]==0,{y[1]==-6,Derivative[1][y][1]==3}},y[x],{x,1,"8"-1}]
 

\[ y(x)\to -\frac {1}{28} (x-1)^7-\frac {1}{20} (x-1)^6+\frac {3}{10} (x-1)^5-\frac {3}{2} (x-1)^3+\frac {3}{2} (x-1)^2+3 (x-1)-6 \]