Internal problem ID [6235]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 18. Power series solutions. 18.9 Indicial Equation with Difference of Roots a
Positive Integer: Logarithmic Case. Exercises page 384
Problem number: 9.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Jacobi]
Solve \begin {gather*} \boxed {x \left (1-x \right ) y^{\prime \prime }+2 \left (1-x \right ) y^{\prime }+2 y=0} \end {gather*} With the expansion point for the power series method at \(x = 1\).
✓ Solution by Maple
Time used: 0.093 (sec). Leaf size: 50
Order:=8; dsolve(x*(1-x)*diff(y(x),x$2)+2*(1-x)*diff(y(x),x)+2*y(x)=0,y(x),type='series',x=1);
\[ y \relax (x ) = \left (1-2 \left (x -1\right )-3 \left (x -1\right )^{2}+2 \left (x -1\right )^{3}-\frac {5}{3} \left (x -1\right )^{4}+\frac {3}{2} \left (x -1\right )^{5}-\frac {7}{5} \left (x -1\right )^{6}+\frac {4}{3} \left (x -1\right )^{7}+\mathrm {O}\left (\left (x -1\right )^{8}\right )\right ) c_{2}+c_{1} \left (x -1\right ) \left (1+\mathrm {O}\left (\left (x -1\right )^{8}\right )\right )+\ln \left (x -1\right ) \left (2 \left (x -1\right )+\mathrm {O}\left (\left (x -1\right )^{8}\right )\right ) c_{2} \]
✓ Solution by Mathematica
Time used: 0.606 (sec). Leaf size: 69
AsymptoticDSolveValue[x*(1-x)*y''[x]+2*(1-x)*y'[x]+2*y[x]==0,y[x],{x,1,7}]
\[ y(x)\to c_2 (x-1)+c_1 \left (\frac {1}{30} \left (-42 (x-1)^6+45 (x-1)^5-50 (x-1)^4+60 (x-1)^3-90 (x-1)^2-90 (x-1)+30\right )+2 (x-1) \log (x-1)\right ) \]