Internal problem ID [6247]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x \left (x -2\right )^{2} y^{\prime \prime }-2 \left (x -2\right ) y^{\prime }+2 y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.028 (sec). Leaf size: 47
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);
\[ y \relax (x ) = \left (\frac {1}{2} x -\frac {1}{8} x^{2}-\frac {1}{48} x^{3}-\frac {1}{192} x^{4}-\frac {1}{640} x^{5}-\frac {1}{1920} x^{6}-\frac {1}{5376} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) c_{2}+\left (1-\frac {1}{2} x +\mathrm {O}\left (x^{8}\right )\right ) \left (\ln \relax (x ) c_{2}+c_{1}\right ) \]
✓ Solution by Mathematica
Time used: 0.011 (sec). Leaf size: 75
AsymptoticDSolveValue[x*(x-2)^2*y''[x]-2*(x-2)*y'[x]+2*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_2 \left (-\frac {x^7}{5376}-\frac {x^6}{1920}-\frac {x^5}{640}-\frac {x^4}{192}-\frac {x^3}{48}-\frac {x^2}{8}+\frac {x}{2}+\left (1-\frac {x}{2}\right ) \log (x)\right )+c_1 \left (1-\frac {x}{2}\right ) \]