54.8.6 problem 8

Internal problem ID [8668]
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 : 8
Date solved : Monday, January 27, 2025 at 04:18:49 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*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.022 (sec). Leaf size: 40

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=0);
 

\[ y = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-\frac {1}{2} x +\operatorname {O}\left (x^{8}\right )\right )+\left (\frac {1}{2} x -\frac {1}{8} x^{2}-\frac {1}{48} x^{3}-\frac {1}{192} x^{4}-\frac {1}{640} x^{5}-\frac {1}{1920} x^{6}-\frac {1}{5376} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \]

✓ Solution by Mathematica

Time used: 0.023 (sec). Leaf size: 75

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

\[ y(x)\to c_2 \left (-\frac {x^7}{5376}-\frac {x^6}{1920}-\frac {x^5}{640}-\frac {x^4}{192}-\frac {x^3}{48}-\frac {x^2}{8}+\frac {x}{2}+\left (1-\frac {x}{2}\right ) \log (x)\right )+c_1 \left (1-\frac {x}{2}\right ) \]