54.3.27 problem 27

Internal problem ID [8583]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 17. Power series solutions. 17.5. Solutions Near an Ordinary Point. Exercises page 355
Problem number : 27
Date solved : Monday, January 27, 2025 at 04:16:47 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

\begin{align*} 2 \end{align*}

✓ Solution by Maple

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

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

\[ y = \left (1-\frac {\left (x -2\right )^{3}}{6}+\frac {\left (x -2\right )^{6}}{180}\right ) y \left (2\right )+\left (x -2-\frac {\left (x -2\right )^{4}}{12}+\frac {\left (x -2\right )^{7}}{504}\right ) y^{\prime }\left (2\right )+O\left (x^{8}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 51

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

\[ y(x)\to c_1 \left (\frac {1}{180} (x-2)^6-\frac {1}{6} (x-2)^3+1\right )+c_2 \left (\frac {1}{504} (x-2)^7-\frac {1}{12} (x-2)^4+x-2\right ) \]