52.1.21 problem 19

Internal problem ID [8238]
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 : 19
Date solved : Monday, January 27, 2025 at 03:46:39 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x -1\right ) y^{\prime \prime }-x y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

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

With initial conditions

\begin{align*} y \left (0\right )&=-2\\ y^{\prime }\left (0\right )&=6 \end{align*}

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 24

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

\[ y = -2+6 x -x^{2}-\frac {1}{3} x^{3}-\frac {1}{12} x^{4}-\frac {1}{60} x^{5}-\frac {1}{360} x^{6}-\frac {1}{2520} x^{7}+\operatorname {O}\left (x^{8}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 48

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

\[ y(x)\to -\frac {x^7}{2520}-\frac {x^6}{360}-\frac {x^5}{60}-\frac {x^4}{12}-\frac {x^3}{3}-x^2+6 x-2 \]