54.8.7 problem 9

Internal problem ID [8669]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.11 Many-Term Recurrence Relations. Exercises page 391
Problem number : 9
Date solved : Monday, January 27, 2025 at 04:18:51 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.043 (sec). Leaf size: 44

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

\[ y = \left (x -2\right ) \left (\left (\ln \left (x -2\right ) c_{2} +c_{1} \right ) \left (1+\operatorname {O}\left (\left (x -2\right )^{8}\right )\right )+\left (-\frac {1}{2} \left (x -2\right )+\frac {1}{8} \left (x -2\right )^{2}-\frac {1}{24} \left (x -2\right )^{3}+\frac {1}{64} \left (x -2\right )^{4}-\frac {1}{160} \left (x -2\right )^{5}+\frac {1}{384} \left (x -2\right )^{6}-\frac {1}{896} \left (x -2\right )^{7}+\operatorname {O}\left (\left (x -2\right )^{8}\right )\right ) c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.031 (sec). Leaf size: 90

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

\[ y(x)\to c_1 (x-2)+c_2 \left (\left (-\frac {1}{896} (x-2)^7+\frac {1}{384} (x-2)^6-\frac {1}{160} (x-2)^5+\frac {1}{64} (x-2)^4-\frac {1}{24} (x-2)^3+\frac {1}{8} (x-2)^2+\frac {2-x}{2}\right ) (x-2)+(x-2) \log (x-2)\right ) \]