54.3.28 problem 28

Internal problem ID [8584]
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 : 28
Date solved : Monday, January 27, 2025 at 04:16:48 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}-2 x +2\right ) y^{\prime \prime }-4 \left (x -1\right ) y^{\prime }+6 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 1 \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 33

Order:=8; 
dsolve((x^2-2*x+2)*diff(y(x),x$2)-4*(x-1)*diff(y(x),x)+6*y(x)=0,y(x),type='series',x=1);
 

\[ y = \frac {\left (-x^{3}+3 x^{2}-2\right ) y^{\prime }\left (1\right )}{3}-3 \left (x^{2}-2 x +\frac {2}{3}\right ) y \left (1\right ) \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 31

AsymptoticDSolveValue[(x^2-2*x+2)*D[y[x],{x,2}]-4*(x-1)*D[y[x],x]+6*y[x]==0,y[x],{x,1,"8"-1}]
 

\[ y(x)\to c_1 \left (1-3 (x-1)^2\right )+c_2 \left (-\frac {1}{3} (x-1)^3+x-1\right ) \]